On local reflection of the properties of graphs with uncountable characteristics (Infinite Combinatorics and Forcing Theory)
18
0
0
全文
(2) 35. are related to generalized and/or modified forms of Fodor‐type Reflection Principle, Strong Changs Conjecture, Rados Conjecture and Galvins Con‐. jecture. As. application of this. an. chromatic number > $\mu$ which \cdot. result. we. show. Further results in this line will be Sakai. reflection theorem. a non. in. on. [11].. the results in Shelah. partially presented covers. Fuchino, Ottenbreit and. [9].. Introduction. 1 For. regular. cardinal $\mu$ and cardinals. \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}\mathrm{o}1}( $\mu$, < $\kap a$, $\lambda$) : coloring. Similarly, \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}\mathrm{h}\mathrm{r} ( $\mu$,. and $\lambda$ with. $\mu$^{+}< $\kappa$\leq $\lambda$. a. ,. let. graph G of cardinality $\lambda$ if G has coloring number subgraph H of G of cardinality < $\kappa$ such that H also. For any. > $\mu$ then there is has. \mathrm{K}. ,. number > $\mu$.. let. < $\kappa$ ). $\lambda$ ):. graph G subgraph. For any. > $\mu$ then there is. a. of. cardinality $\lambda$ if G has chromatic number H of G of cardinality < $\kappa$ such that H also ,. has chromatic number > $\mu$. In this note. Theorem 1.1. (1.1). shall. we. give. Suppose. we. Stress is put here. $\kappa$^{< $\mu$}= $\kappa$ hold. of the are. does not hold then. have. on. following. Theorem:. cardinals such that. $\kappa$^{< $\mu$}= $\kappa$ and $\lambda$<$\kappa$^{+ $\omega$}.. Note that the condition. Kónigs Theorem,. proof. that $\mu$_{f} $\kappa$, $\lambda$. $\mu$^{< $\mu$}= $\mu$, $\mu$^{+}< $\kappa$,. If \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}\circ 1}( $\mu$, < $\kap a,\ \lambda$^{ $\mu$}). a. the. \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}\mathrm{h}\mathrm{r} ( $\mu$, < $\kap a,\ \lambda$^{ $\mu$}). $\mu$^{< $\mu$}= $\mu$ implies that $\mu$ is a regular $\kappa$^{cf( $\kappa$)}> $\kappa$ for any cardinal $\kappa$. the condition $\lambda$<$\kappa$^{+ $\omega$}. the assertion of Theorem 1.1. For. more see. can. since, by. $\mu$^{< $\mu$}= $\mu$ and dropped from. also be. [8].. Further results in this line for uncountable $\mu$ will be. presented. in. Fuchino,. [9].. Preliminary. 2. In this note V. cardinal. where $\mu$ is uncountable. For $\mu$= $\omega$,. cases. automatically and. OttenUreit and Sakai. does not hold.. are. not. (and in. going. the further article. to denote. a. generic. [9]. in. preparation). set and the. ground. the Roman letters G and. model. as. it is usual the.
(3) 36. theory. in set. case. We shall. but rather. they. are. used to denote. v and \mathrm{G} instead for the. a. graph and its set of vertices. a generic set respectively.. model and. ground graph as a structure of the form G= \langle V, \mathcal{E} } where \mathcal{E}\underline{\subseteq}V^{2} and \mathcal{E} is thought to be the symmetrical and non reflective binary relation representing the adjacency of the graph G We often identify G with the underlying set V of G and even write G=\{G, \mathcal{E}\} For X\subseteq G and p\in G (i.e. for X\subseteq V and p\in V ), we use. We consider. a. .. .. denote. \mathcal{E}_{X}^{p}=\{q\in X : q\mathcal{E}p\}.. (2.1). Recall that the. coloring number of a graph G=\{G, \mathcal{E} ). is defined. as. the minimal. cardinal $\mu$ with the property that. (2.2). there is. a. \mathrm{w}\mathrm{e}\mathrm{U}- ordering \triangleleft \mathrm{o}\mathrm{n}G such that for any p\in. segment below p with respect we. The. have. to the. ordering. |\mathcal{E}_{I_{\mathrm{p} ^{\triangle ft} ^{p}|< $\mu$.. G denoting the initial ). \triangleleft \mathrm{b}\mathrm{y}I_{p}^{\triangleleft}=\{q\in G : q\triangleleft p\},. coloring number of a graph G is denoted by col (G) a graph G \{V, \mathcal{E}\} and X \subseteq V, G \mathrm{} X denotes the induced subgraph .. For. =. {X, \mathcal{E}\cap X^{2}\rangle. of G. is said to be. a. |X_{ $\alpha$}|. for all $\alpha$< $\delta$ and. <. |X|. .. For. filtration. a. set X. a. $\delta$\rangle of subsets of X continuously increasing (with respect to \subseteq ),. sequence \mathcal{F}=. of X if \mathcal{F} is. \displaystyle \bigcup_{ $\alpha$< $\delta$}X_{ $\alpha$}=X. .. \{X_{ $\alpha$}. Note. :. that,. $\alpha$. <. for any set X ,. we. have. a. strictly increasing filtration of X of length $\delta$=cf(|X For a graph G= (V, \mathcal{E}\rangle a sequence \{G_{ $\alpha$} : $\alpha$ < $\delta$\} of induced subgraphs of G with G_{ $\alpha$} G \mathrm{r}V_{ $\alpha$} for $\alpha$ < $\delta$ is said to be a filtration of G if \langle V_{ $\alpha$} : $\alpha$ < $\delta$ } is a =. filtration of V. The. following. Lemma is. proved easily by (simultaneous). induction. the. on. car‐. dinality of G : Lemma 2.1. (Erdó\prime\mathrm{s}. and. (2) of. For any. Hajnal [3]. see. also. [5]) (1) If $\mu$=\mathrm{c}\mathrm{o}1(G). of |G| graph G co1 (G)\leq $\mu$ if and only if there. well‐orden ng\triangleleft on G. order type. there is. a. witnessing this.. ,. \dot{u}. a. filtration \langle G_{ $\alpha$}. :. $\alpha$< $\delta$ }. G such that. (2.3). col (G_{ $\alpha$})\leq $\mu$ and. (2.4). |\mathcal{E}_{G_{ $\alpha$} ^{p}|< $\mu$ for. all. p\in G_{ $\alpha$+1}\backslash G_{ $\alpha$}. for all $\alpha$< $\delta$. Recall. that, for. \square a. graph G. =. \{G, \mathcal{E}\}. the minimal cardinal $\mu$ such that G. can. ,. be. the chromatic number. partitioned. \mathrm{c}\mathrm{h}\mathrm{r}(G). into $\mu$ many. of G is. pairwise. non. adjacent (i.e. independent) subgraphs. . Suppose that inv i\mathcal{S} one of col or chr. If \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{i}\mathrm{n}\mathrm{v} ( $\mu$, < $\kap a$, $\lambda$) holds, $\kappa$'\geq $\kappa$ and $\lambda$'\leq $\lambda$ then REFL j\mathrm{n}\mathrm{v}( $\mu$, <$\kappa$', $\lambda$') holds.. Lemma 2.2. ,.
(4) 37. Suppose that \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{i}$\iota$/\mathrm{v} ( $\mu$, < $\kappa$', $\lambda$') does not hold and let G= \langle G, \mathcal{E} } be graph cardinality $\lambda$' which is a witness of the failure of REFL \mathrm{i}\mathrm{n}\mathrm{v}( $\mu$, < $\kappa$', $\lambda$. Proof.. of. a. (G)> $\mu$. Thus inv. Let G' be. but inv. (G_{0})\leq $\mu$. G_{0}\in[G]^{<$\kappa$'}. for all. G\cup G' Then cardinality disjoint from G and let G_{1} the graph G_{1} \{G_{1}, \mathcal{E}\rangle is of cardinality $\lambda$ inv (G_{1}) > $\mu$ but inv (G_{0}) \leq $\mu$ for all G_{0}\in [G_{1}]^{<$\kappa$'} (and hence this holds for all G_{0}\in [G_{1}]^{< $\kappa$} ). Thus G_{1} is a witness of a. set of. $\lambda$. =. =. the failure of. .. .. \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{i}\mathrm{n}\mathrm{v} ( $\mu$, < $\kap a$, $\lambda$). .. \square. Refiection. 3. properties. Fodor‐type For. Reflection. $\mu$, $\kappa$, $\lambda$ with. regular cardinals. related to. (Lemma 2.2). generalized. Principles. $\mu$^{+}< $\kappa$\leq $\lambda$. let. ,. \mathrm{F}\mathrm{R}\mathrm{P}( $\mu$, < $\kappa$, $\lambda$). be the. following. assertion:. \mathrm{F}\mathrm{R}\mathrm{P}( $\mu$, < $\kappa$, $\lambda$) :. For any. E_{> $\mu$}^{ $\lambda$}\cap E_{< $\kap a$}^{ $\lambda$}. stationary S \underline{\subset q}. E_{$\mu$}^{$\lambda$}. Using this notation, be formulated. the. S. \rightarrow. [ $\lambda$]^{ $\mu$}. there is. is. stationary. in. \{x. an. $\alpha$^{*} \in. [$\alpha$^{*}]^{ $\mu$}. \in. :. [$\alpha$^{*}]^{ $\mu$}.. Fodor‐type Reflection principle (FRP) introduced. in. [4]. as. \mathrm{F}\mathrm{R}\mathrm{P} (\aleph_{0}\text{）}<\aleph_{2}, $\lambda$) holds for all. FRP:. :. such that $\alpha$^{*} is closed with respect to g and. \displaystyle \sup(s)\in S, g(\displaystyle \sup(x))\cap\sup(x)\subseteq x\}. can. and g. regular uncountable. $\lambda$.. equivalent (over ZFC) to many known mathematical reflec‐ one saying that a locally compact Hausdorff space X is metrizable if and only if all subspace of X of size \leq \aleph_{1} are metrizable. FRP also implies many interesting consequences like SCH while it does not restrict the size FRP is known to be. tion statements like the. of the continuum unlike many other reflection are. \aleph_{2}. going. to discuss below which. or even. imply. principles. Conjecture we than or equal to. like Rados. that the continuum is less. CH.. following result of Hiroshi Sakai, this principle cannot be consistently generalized by taking an uncountable $\mu$ in place of \aleph_{0} in \mathrm{F}\mathrm{R}\mathrm{P}(\aleph_{0}, <\aleph_{2}, $\lambda$) A cardinal $\kappa$ is said to be $\lambda$ ‐inaccessible if $\mu$^{ $\lambda$}< $\kappa$ holds for all $\mu$< $\kappa$ Similarly By. the. .. .. we. shall also say that. Theorem 3.1. regular. (H.. $\kappa$. Sakai. This delimitation set tions. on. [9]). Let $\lambda$ be. $\mu$^{< $\lambda$}< $\kappa$. holds for all $\mu$< $\kappa$.. singular cardinal, and let $\mu$ and $\mu$^{+} Suppose that every regular cardinal \mathrm{v} cf( $\lambda$) ‐inaccessible. Then \mathrm{F}\mathrm{R}\mathrm{P}( $\mu$, < $\kappa,\ \lambda$^{+}) fails.. cardinals with. $\mu$< $\nu$< $\kappa$ is. is < $\lambda$ ‐inaccessible if. cardinals in. <. $\kappa$. a. \leq $\lambda$. by the theorem above explains the the following Proposition.. $\lambda$ be with. releance of the condi‐.
(5) 38. In a. spite of Theorem 3.1,. reasonable. generalization. modify the property \mathrm{F}\mathrm{R}\mathrm{P}( $\mu$, < $\kappa$, $\lambda$) to obtain higher cardinals. This will Ue discussed in. we can. of FRP for. [9]. 3.2 For any cardinals $\mu$,. Proposition. and $\lambda$^{*} such that. $\kappa$. $\mu$^{+}< $\kappa$\leq$\lambda$^{*}<$\kappa$^{+ $\omega$},. (3.1). if \mathrm{F}\mathrm{R}\mathrm{P}( $\mu$, < $\kappa$, $\lambda$) Proof.. holds. for all $\lambda$<$\lambda$^{*} then \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}\mathrm{o} |( $\mu$, < $\kap a$, $\lambda$) holds for all $\lambda$<$\lambda$^{*}.. By induction of. $\lambda$^{*}. If $\lambda$^{*} \leq $\kappa$ then \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}\mathrm{o} | ( $\mu$\text{）}< $\kappa$, $\lambda$). .. trivially. holds for. all $\lambda$<$\lambda$^{*}.. Proposition holds for all $\kappa$\leq$\lambda$_{0}^{*}<$\lambda$^{*} and show that Proposition also holds for $\lambda$^{*} By (3.1), there is $\lambda$_{0} < $\lambda$^{*} such that ($\lambda$_{0})^{+} Thus it is enough to show that \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}\mathrm{o} | ( $\mu$, < $\kappa$ $\lambda$_{0} ) holds. We. assume. that the. the $\lambda$^{*}.. =. .. ). graph G of cardinality $\lambda$_{0} such that co1 (G) > $\mu$ but all subgraphs H of G of cardinality < $\kappa$ have coloring number \leq $\mu$ Without loss of generality G=\{$\lambda$_{0}, \mathcal{E}\} for some adjacency relation \mathcal{E}. Let \langle$\eta$_{ $\alpha$} : $\alpha$<$\lambda$_{0}\} be a continuously and strictly increasing sequence of ordinals cofinal in $\lambda$_{0} and $\xi$_{ $\alpha$}\in$\eta$_{ $\alpha$+1}\backslash $\eta$_{ $\alpha$} for $\alpha$<$\lambda$_{0} are such that, for all $\alpha$<$\lambda$_{0} Suppose. that this is not the. case.. Then there is. a. .. ,. (3.2). If. |\mathcal{E}_{$\eta$_{ $\alpha$} ^{ $\xi$}|\geq $\mu$ for. By induction hypothesis. (3.3). some. we. $\xi$\in$\lambda$_{0}\backslash $\eta$_{ $\alpha$}. ,. then. |\mathcal{E}_{$\eta$_{ $\alpha$}^{ $\alpha$} ^{ $\xi$}|\geq $\mu$.. have co1 (G($\eta$_{ $\alpha$})\leq $\mu$ for all. S=\{ $\alpha$<$\lambda$_{0} : |\mathcal{E}_{$\eta$_{ $\alpha$}^{ $\alpha$}}^{ $\xi$}|\geq $\mu$\}. is. $\alpha$<$\lambda$_{0}. .. Thus. stationary. (since otherwise we would obtain col (G)\leq $\mu$ by Lemma 2.1. assumption on G ).. This is. a. contradiction. to the. S_{1}=S\cap E_{$\mu$^{\mathrm{O} }^{ $\lambda$}. Claim 3.2.1. \vdash Suppose. that S_{1}. ts. stationary.. were non. stationary. Then,. at least. one. of. S_{0}. =. S\cup E_{< $\mu$}^{$\lambda$_{0}. stationary. Suppose that i\in { 0 2} is such that S_{i} is S_{2}=S\cup E_{> $\mu$}^{$\lambda$_{0} Then for each $\alpha$\in S_{i} there is $\nu$_{ $\alpha$}<$\eta$_{ $\alpha$} such that |\mathcal{E}_{$\nu$_{ $\alpha$}^{ $\alpha$} ^{ $\xi$}\geq $\mu$| By Fodors stationary. Lemma, there is a stationary S_{4}\subseteq S_{i} and $\nu$^{*}<$\lambda$_{0} such that \mathrm{v}_{ $\alpha$}=$\nu$^{*} for all $\alpha$\in S_{4}. and. would be. ). .. It follows that. E_{$\mu$^{0} ^{ $\lambda$}\displaystyle \backslash \sup(\mathrm{v}^{*})\subseteq S. .. This is. inclusion is stationary and it is thus For each. $\alpha$. \mathrm{E}. S_{1} let ,. s_{ $\alpha$} \in. g( $\alpha$)=s_{ $\alpha$}\cup\{$\xi$_{ $\alpha$}\} for $\alpha$\in S_{1}. By \mathrm{F}\mathrm{R}\mathrm{P}( $\mu$, < $\kappa,\ \lambda$_{0}) there ,. is. a. [\mathcal{E}_{$\eta$_{$\alpha$}^{$\alpha$}^{$\xi$}]^{$\mu$}. a. contradiction since the left side of the. subset of and let g. S_{1}=S\cap E_{$\mu$^{0} ^{ $\lambda$}. :. S_{1}. \rightarrow. [$\lambda$_{0}]^{ $\mu$}. $\alpha$^{*}\inE_{>$\mu$}^{$\lambda$_{\mathrm{O} \capE_{<$\kap a$}^{$\lambda$_{\mathrm{O} such that $\alpha$^{*}. :. \displaystyle \sup(x)\in S_{1}. and. (Claim. 3.. be the defined. 2.1\rangle. by. is closed with respect. to g and. { x\in[$\alpha$^{*}]^{ $\mu$}. \dashv. g(\displaystyle \sup(x))\cap\sup(x)\subseteq x}.
(6) 39. is. stationary. It follows that there. cf($\alpha$^{*})\}. such that each of. S= is a. { $\xi$<cf($\alpha$^{*}). stationary.. But. is. I_{ $\xi$}, $\xi$<cf($\alpha$^{*}). \displaystyle \sup(I_{ $\xi$})\in S_{1}. :. then, by. Lemma. I\in. an. 2.1,. [$\alpha$^{*}]^{cf($\alpha$^{*})}. with. a. filtration. and. Reflection. must conclude co1. we. (G (I)> $\mu$ 0. principles. $\xi$<. :. g(\displaystyle \sup(I_{ $\xi$}))\cap\sup(I_{ $\xi$})\subseteq I_{ $\xi$} }. contradiction to the choice of G.. 4. \{I_{ $\xi$}. is closed with respect to g and. related to. a. .. This is. (proposition 3.2). variant of. Strong Changs Conjecture (4.1). Let $\theta$ be. regular cardinal large enough (compared with $\lambda$ below). \mathcal{M}=(\mathcal{H}( $\theta$)\text{）\in}, \subset\rangle where \sqsubset\mathrm{i}\mathrm{s} a fixed well‐ordering of \mathcal{H}( $\theta$). Let. a. .. The. well‐ordering. \subset is included in the structure. \mathcal{M} here because of the built‐in. Skolem functions it introduces.. following principle \mathrm{C}\mathrm{C}^{\downar ow} ( $\mu$, < $\kappa$, $\lambda$) is a generalized version of a principle considered in [8]. These principles are inspired by a variant of Strong Changs Conjecture in Doebler [2] \dot{ex）} the Strong Changs Conjecture in its original form was The. introduced in Todorčevič For. regular cardinal. a. \mathrm{C}\mathrm{C}^{\downar ow}( $\mu$, < $\kap a$, $\lambda$) \mathrm{C}\mathrm{C}^{\downar ow} ( $\mu$,. < $\kappa$ ). [15]. $\mu$ and cardinals $\kappa$, $\lambda$ with. be the assertion defined. $\lambda$ ):. For any. [M]^{< $\mu$}\subseteq M (4.2). ,. we. M \in. 4.1. (4.4). $\mu$^{< $\mu$}= $\mu$ ;. (4.5). $\mu$^{+}< $\kappa$\leq $\lambda$. (4.6). $\lambda$. (4.7). are. Assume .. $\mu$^{+}< $\kappa$\leq $\lambda$. ,. let. \mathcal{M},. $\mu$ \subseteq. M $\mu$, $\kappa$, $\lambda$ \in. M and. $\mu$,. $\kap a$_{f} $\lambda$. and. $\alpha$^{*}\in $\lambda$\backslash $\alpha$ $\alpha$^{*}=\displaystyle \min( $\lambda$\cap M^{*}\backslash \sup( $\lambda$\cap M. cardinals such that. are. regular; and. is< $\mu$ ‐inaccessible.. \mathrm{C}\mathrm{C}^{\downar ow}( $\mu$, < $\kap a$, $\lambda$). S\rightar ow[ $\lambda$]^{ $\mu$}. with M \prec. $\mu$<cf($\alpha$^{*})< $\kappa$. Suppose that. $\mu$ and $\lambda$. Proof.. and. for any $\alpha$\in $\lambda$ , there is M^{*} with M\prec M^{*} \prec \mathcal{M} and. (4.3). Then. $\mu$^{< $\mu$}= $\mu$. follows:. have that. such that. Proposition. [\mathcal{M}]^{ $\mu$}. as. Let. implies. \mathrm{C}\mathrm{C}^{\downar ow} ( $\mu$, < Ki, $\lambda$) and. M\in[\mathcal{M}]^{ $\mu$}. M\prec \mathcal{M} ;. \mathrm{F}\mathrm{R}\mathrm{P}( $\mu$, < $\kap a$, $\lambda$). .. suppose that S. be such that. \subseteq. E_{$\mu$}^{$\lambda$}. is. stationary. and g. :.
(7) 40. (4.8). $\mu$ ) $\kappa$,. (4.9). $\mu$\subseteq M ;. $\lambda$, S, g\in M ;. (4.10). [M]^{< $\mu$}\subseteq M ;. (4.11). M is closed with respect to g ;. (4.12). \displaystyle \sup( $\lambda$\cap M)\in S. Note that there is such. g(\displaystyle \sup(M))\cap\sup(M)\subseteq M.. and M. an. by (4.4). and. (4.6). By \mathrm{C}\mathrm{C}^{\downar ow}( $\mu$\text{）}< $\kap a$, $\lambda$). there. are. $\alpha$^{*}\in $\lambda$. and M^{*}\prec \mathcal{M} such that. (4.13). M\prec M^{*} ;. (4.14). $\mu$<cf($\alpha$^{*})< $\kappa$ ;. (4.15). $\alpha$^{*}=\displaystyle \min( $\lambda$\cap M^{*}\backslash \sup( $\lambda$\cap M. and. We show that this $\alpha$^{*} witnesses \mathrm{F}\mathrm{R}\mathrm{P} ( $\mu$,. < $\kappa$ ). $\lambda$ ) for. our. S and g.. $\alpha$^{*} is closed with respect to g since it is closed with respect to g in M^{*}. Thus it is. (4.16) is. enough. Z=. { x\in[$\alpha$^{*}]. :. \displaystyle \sup(x)\in S. stationary. By elementarity,. sets of. [$\alpha$^{*}]^{ $\mu$}. Suppose. by (4.15).. to show that. it is. and. g(\displaystyle \sup(x))\cap\sup(x)\subseteq x}. enough. to show that Z intersects with all club. in M^{*}.. that C \in M^{*} is. a. club subset of. [$\alpha$^{*}]^{ $\mu$}. and let h \in M^{*} be such that. h:^{ $\omega$>}$\alpha$^{*}\rightarrow$\alpha$^{*} and. (4.17) Then. C\supseteq C_{h}= { x\in[$\alpha$^{*}]^{ $\mu$} we. (4.18). :. $\mu$\subseteq x and. x. is closed with respect to h }.. have. $\alpha$^{*}\cap M\in Z\cap C_{h}. [ $\alpha$^{*}\cap M\in Z by (4.12). and. $\alpha$^{*}\cap M\in C_{h} by (4.9)]. Thus Z\cap C\neq\emptyset.. 0. (Proposition 4.1). mapping f : $\lambda$\rightar ow $\lambda$ is said to be regressive if f( $\alpha$)< $\alpha$ holds for all $\alpha$< $\lambda$ We denote with $\lambda$\downar ow$\lambda$ the set { f\in $\lambda \lambda$ : f is regressive}. For. a. regular cardinal. $\lambda$ ,. a. .. The game. G_{ $\mu$}^{\downar ow}(< $\kap a$, $\lambda$). is. G_{ $\mu$}^{\downar ow}(< $\kap a$, $\lambda$) a. sequence. for. Players I and II is defined of length $\mu$ of the form:. as. follows: A match in. ( $\xi$< $\mu$). \mathfrak{M} : II wins in. (4.19). a. match \mathfrak{M} of. G_{ $\mu$}^{\downar ow}(< $\kap a$, $\lambda$). B_{\mathfrak{M}}= { $\alpha$\in E_{> $\mu$}^{ $\lambda$}\cap E_{< $\kap a$}^{ $\lambda$}. :. as. above if. f_{ $\xi$}( $\alpha$)<\displaystyle \sup\{$\delta$_{i}. :. i< $\mu$\}. for all. $\xi$< $\mu$ }. is unbounded in $\lambda$.. Let the. us. denote with. player. I. \mathrm{W}\mathrm{S}_{I}(G_{ $\mu$}^{\downar ow}(< $\kap a$, $\lambda$) ( \mathrm{W}\mathrm{S}_{I }(G_{ $\mu$}^{\downar ow} ( < $\kappa$, $\lambda$. (the player II, resp.). has. a. winning strategy. resp.). the assertion. in the game. G_{ $\mu$}^{\downar ow}(< $\kap a$, $\lambda$). ..
(8) 41. Proposition. Suppose. 4.2. that. $\mu$^{< $\mu$}= $\mu$. (4.20). Then, for any cardinals $\kappa$, $\lambda$ with $\mu$^{+} only if \mathrm{C}\mathrm{C}^{\downar ow}( $\mu$, < $\kap a$, $\lambda$) holds.. holds.. and. Proof.. Suppose. \mathrm{W}\mathrm{S}_{I }(G_{ $\mu$}^{\downar ow}(< $\kap a$, $\lambda$). first that. $\kappa$\leq $\lambda$, \mathrm{W}\mathrm{S}_{I } (G_{ $\mu$}^{\downar ow} (< $\kap a$, $\lambda$). <. holds. Let \mathcal{M} be defined. as. hold\mathcal{S}. in. if. (4.1). and let M\prec \mathcal{M} be such that. (4.21). |M|= $\mu$ ;. (4.22). [M]^{< $\mu$}\subseteq M ;. (4.23). $\mu$, $\kappa$, $\lambda$\in M and. $\mu$\subseteq M.. by (4.20). wining strategy of. Note that there is such M Let. $\sigma$. M be. \in. $\xi$< $\mu$). be. (4.24). \{f_{ $\xi$}, $\delta$_{$\xi$}. :. (4.25). II. (4.26). \{f_{ $\xi$}. \{f_{ $\xi$}, $\delta$_{$\xi$}. :. Note that. a. $\xi$< $\gamma$\rangle\in M. plays according :. $\xi$< $\mu$\rangle. (4.24). is. G_{ $\mu$}^{\downar ow}(< $\kap a$, $\lambda$). match in. a. player. II in. such that. G_{ $\mu$}^{\downar ow}(< $\kap a$, $\lambda$). Let \mathfrak{M}. .. =. for all $\gamma$< $\mu$ ;. to. in \mathfrak{M} ;. $\sigma$. enumerates. possible. the. $\lambda$\downarrow $\lambda$\cap M.. because of. (4.22).. Since II wins in the match \mathfrak{M} , there is $\alpha$^{*} such that. (4.27). \displaystyle \sup( $\lambda$\cap M)<$\alpha$^{*} ;. (4.28). $\alpha$^{*}\in E_{> $\mu$}^{ $\lambda$}\cap E_{< $\kap a$}^{ $\lambda$}. (4.29). f_{ $\xi$}($\alpha$^{*})<\displaystyle \sup\{$\delta$_{ $\xi$} : $\xi$< $\mu$\}\leq\sup( $\lambda$\cap M). and. Since all Skolem function a. f. in M with. regressive function from $\lambda$ to $\lambda$. M^{*}=sk_{\mathcal{M}}(M\cup\{$\alpha$^{*}\}). is. as. are. .. parameters from M such that f [ $\lambda$ is. among. f_{ $\xi$}, $\xi$. <. $\mu$ , it is. readily. seen. that. desired.. Suppose \mathrm{C}\mathrm{C}^{\downar ow}( $\mu$, < $\kap a$, $\lambda$) holds. In a match \mathfrak{M} the player II can choose a continuously increasing sequence \{M_{ $\xi$} : $\xi$ < $\delta$\} of elementary submodels of \mathcal{M} such that, for all $\xi$< $\mu$, now. that. ,. (4.30). |M_{ $\xi$+1}|= $\mu$ ;. (4.31). [M_{ $\xi$+1}]^{< $\delta$}\underline{\subseteq}M_{ $\xi$+1}. (4.32). $\delta$_{ $\xi$}=\displaystyle \sup( $\lambda$\cap M_{ $\xi$}). Then. .. M=\displaystyle \bigcup_{ $\xi$< $\mu$}M_{ $\xi$} is an elementary submodel of \mathcal{M} of cardinality $\mu$ with [M]^{< $\mu$}\subseteq. M and \displaystyle \sup (\{$\delta$_{ $\xi$} : $\xi$ < $\mu$\}). \mathrm{C}\mathrm{C}^{\downar ow} ( $\mu$,. < $\kappa$ ). =. $\lambda$ ) for this M is in. Let $\theta$ and \mathcal{M} be. as. in. \displaystyle \sup( $\lambda$\cap M). .. Thus each $\alpha$^{*}. B_{\mathfrak{M} .. (4.1).. For. as. in the definition of 0. M\in[\mathcal{M}]^{ $\mu$}. such that. (proposition 4.2).
(9) 42. (4.33). M\prec \mathcal{M} ;. (4.34). $\mu$\subseteq M,. (4.35). [M]^{< $\mu$}\subseteq M,. $\mu$ , $\kappa$, $\lambda$\in M ; and. let. D_{M}= { $\alpha$\in E_{> $\mu$}^{ $\lambda$}\cap E_{< $\kap a$}^{ $\lambda$}. (4.36) Clearly Let. (4.37). f( $\alpha$)<\displaystyle \sup( $\lambda$\cap M). D_{M}\displaystyle \supseteq\sup( $\lambda$\cap M)\cap(E_{> $\mu$}^{ $\lambda$}\cap E_{< $\kappa$}^{ $\lambda$}). have. we. :. \mathcal{B}=\{M\in[\mathcal{M}]^{ $\mu$}. ,. Lemma 4.3 $\alpha$^{*}< $\lambda$ is. The. upper bound. an. \mathcal{S}omef\in $\lambda$\downarrow $\lambda$\cap M. there is. characterization of. following. tant role in the next section.. Suppose. Lemma 4.4. \mathrm{W}\mathrm{S}_{I }(G_{ $\mu$}^{\downar ow}(< $\kap a$, $\lambda$) Proof.. If. player non‐stationary. (4.34). cf( $\lambda$) \leq $\mu$ II. and. Thus. (4.35). we. By. the. (4.26).. (4.38) to. $\mu$^{+}. <. choose her. f( $\alpha$)\displaystyle \geq\sup( $\lambda$\cap M). $\kappa$\leq $\lambda$. .. is. since there. going. are. $\delta$_{ $\xi$}, $\xi$< $\mu$. \mathrm{O}. play. an. impor‐. any cardinal. many. $\lambda$,. [\mathcal{M}]^{ $\mu$}.. \mathrm{W}\mathrm{S}_{I }(G_{ $\mu$}^{\downar ow}(< $\kap a$, $\lambda$). such that. end‐segment. $\alpha$\in(E_{> $\mu$}^{ $\lambda$}\cap. .. to. Then, for. if \mathcal{B} is non‐stationary in. moves. any. holds. \{$\delta$_{ $\xi$} : $\xi$< $\mu$\} M\in[\mathcal{M}]^{ $\mu$} with (4.33),. is cofinal. cf( $\lambda$) > $\mu$ Suppose first that \mathrm{W}\mathrm{S}_{I }(G_{ $\mu$}^{\downar ow}(< $\kap a$, \mathrm{A}) holds. C=\{M\in[\mathcal{M}]^{ $\mu$} : M\prec \mathcal{M}\} is disjoint from \mathcal{B} Suppose M\in \mathcal{B}\cap C. .. .. \langle f_{ $\xi$}, $\delta$_{$\xi$}. Since the. there is. a. $\xi$ < $\mu$\rangle be a match player II wins in \mathfrak{M}. By (4.26). :. player II in G_{ $\mu$}^{\downar ow}(< $\kap a$, $\lambda$) satisfying (4.24), (4.25) and. wining strategy $\sigma$\in M of the. :. { $\alpha$\in E_{> $\mu$}^{ $\lambda$}\cap E_{< $\kap a$}^{ $\lambda$}. is unbounded.. only if, for. \mathrm{W}\mathrm{S}_{I }(G_{ $\mu$}^{\downar ow}(< $\kap a$, $\lambda$). = $\mu$ and. only. and. this is clear. Under this condition. assume. assumption,. Let \mathfrak{M}=. if. and. of D_{M} if. such that. D_{M}.. such that $\lambda$\cap M is cofinal in $\lambda$.. may. We show that. can. $\mu$^{< $\mu$}. that. holds. since the in $\lambda$. \mathcal{B} is. and D_{M} is bounded in $\lambda$ }.. is immediate from the definition of. following. E_{< $\kap a$}^{ $\lambda$})\backslash $\alpha$^{*}. .. M\prec \mathcal{M},. :. M\models(4.34) (4.35) The. f\in $\lambda$\downarrow $\lambda$\cap M }.. for all. in. G_{ $\mu$}^{\downar ow}(< $\kap a$, $\lambda$). f_{ $\xi$}( $\alpha$)<\displaystyle \sup(\{$\delta$_{i}. it follows that. :. i< $\xi$\}). .. for all. $\xi$< $\mu$ }. D_{M} is unbounded. This is. a. contradiction. club. disjoint from. M\in \mathcal{B}.. Suppose \mathcal{B} We may .. (4.39). now. that \mathcal{B} is non‐stationary. Let C\subseteq. assume. that. M\prec \mathcal{M} holds for all M\in C.. [\mathcal{M}]^{ $\mu$}. be. a.
(10) 43. In. a. \langle f_{ $\xi$}, $\delta$_{$\xi$}. match \mathfrak{M}=. such. a. way. :. $\xi$< $\mu$\rangle. ,. the. player. that, along a continuously increasing. should build. $\mu$\subseteq M_{0},. (4.41). \{f_{ $\eta$} : $\eta$\leq $\xi$\}\subseteq M_{ $\xi$+1} for all $\xi$< $\mu$ ; [M_{ $\xi$}]^{< $\mu$}\subseteq M_{ $\xi$+1} for all $\xi$< $\mu$ ; and $\delta$_{ $\xi$}=\displaystyle \sup( $\lambda$\cap M_{ $\xi$}) for all $\xi$< $\mu$.. (4.43). Since C is. (4.42),. a. $\mu$, $\kappa$,. club M. M satisfies. (4.44). choose her. moves. \{M_{ $\xi$} : $\xi$< $\mu$\}. =. \displaystyle \bigcup_{ $\xi$< $\mu$}M_{ $\xi$}. (4.33), (4.34). C and hence M. \in. (4.35).. and. \not\in \mathcal{B}. By (4.39), (4.40). .. In the. following. \{s\in T : \mathcal{S}<T\} single. a. it follows from. properties. (4.44). that the. we assume. that. is well‐ordered. root.. By. player. related to. a. II wins (Lemma. 4 .4). generaliza‐. Conjecture a. by. tree is. < $\tau$. .. In. partial ordering T=\{T, <T\rangle such that particular, we do not assume that a tree. a. this convention any subset of. a. tree T. can. be considered. as. subtree of T. A tree T is said to be $\mu$ ‐special if T. ( $\delta$ \leq $\mu$) antichain). $\alpha$. and. Thus. \square. tion of Rados. a. in. D_{M} is unbounded.. Reflection. has. ). of C which. and such that. in all such matches \mathcal{M}.. 5. M_{ $\xi$}. $\delta$_{$\xi$} $\xi$< $\mu$. $\lambda$\in M_{0} ;. \{f_{ $\xi$} : $\xi$< $\mu$\}\underline{\subseteq}M by (4.41),. Since. can. sequence. (4.40) (4.42). II. with her moves, she also chooses elements. <. $\delta$. The. can. be. into \leq $\mu$ subsets. incomparable (i.e. each T_{ $\alpha$}. such that each T_{ $\alpha$} is pairwise. following reflection property. partitioned. is related to. a. T_{ $\alpha$},. is. an. generalization of Rados Con‐. jecture: RC ( $\mu$, < $\kappa$, $\lambda$) : a. Note. For any tree T of. subtree T' of T of size. that, using. this. \mathrm{R}\mathrm{C}(\aleph_{0}, <\aleph_{2}, $\lambda$). RC: The. ,. if T is not $\mu$‐special, then there is. such that T' is not $\mu$‐special.. notation, Rados Conjecture (RC). can. be reformulated. 5.4 for $\mu$. \aleph_{0} together with Proposition 4.2 for of Proposition 4.1 for $\mu$= \aleph_{0} proves that Rados =. slight extension Conjecture implies Fodor‐type Reflection Principle (see Fuchino, Sakai, Torres $\mu$=. Usuba. as:. holds for all cardinal $\lambda$.. following Proposition. \aleph_{0} and. cardinality $\lambda$ < $\kappa$. a. and. [8]).. In contrast to FRP the. uncountable cardinals:. straightforward generalization of Rado \mathrm{s} Conjecture. to.
(11) 44. \mathrm{R}\mathrm{C} ( $\mu$\text{）}<$\mu$^{++}, $\lambda$) holds for all cardinal $\lambda$. \mathrm{R}\mathrm{C}_{$\mu$} :. is consistent. We shall discuss. As mentioned before RC. \leq\aleph_{2} Starting from. a. .. of. 2^{\aleph_{0} =\aleph_{1} Let. us. begin. with. some. that the. implies. compact cardinal. super. 2^{\aleph_{0} =\aleph_{2} (see. or. Todorčevič. tools. we. \leq $\mu$. again. [13], [15]).. need for the. said to be \leq $\mu$‐Baire if the intersection is. generalizaion in [9]. cardinality of the continuum to be we can force RC together with each. about this. more. proof of Proposition. 5.4. A tree is. \cap D of any open dense subsets of cardinality. open dense where D\underline{\subseteq}T is said to be open dense if it is. and for any t\in T there is t'\in T with t<_{T}t. . (i.e.. D is open. upward closed dense in the forcing. by putting upside down). following is easy to prove:. poset obtained The. T. (1). Lemma 5.1. Let T be. a. tree without maximal elements.. If T is\leq. $\mu$ ‐Baire. then T is not $\mu$ ‐special.. (2) If a. tree T is. (3) Any a. f(t)< $\tau$ t. subtree. case. f special. For. :. [12]. given. Theorem 5.2 that. T_{0} of. holds for all. Todorčevič. general. T is $\mu$ ‐special.. if T has $\mu$ ‐special.. tree T is not $\mu$ ‐special. of height >$\mu$^{+}. any tree. For. of height <$\mu$^{+} then is not a. tree T ,. proves the can. (Pressing. T\rightarrow T is. branch. (5.1) Then. (5.2). .. In. particular,. T_{0}\rightarrow T is said. to be. regressive. if. is not minimal in T.. following Theorem only for the case $\mu$=\aleph_{0} proved with exactly the same proof.. but the. be. and. Trees, Todorčevič [12]) Suppose for all t \in T then T is $\mu$-. f^{-1\prime\prime}\{t\}. is $\mu$ ‐special. \square a. tree T , let. Lim(T)= { t\in T. :. ht_{T}(t). Corollary 5.3 Suppose that f : Lim(T)\rightarrow T for all t\in T then T is $\mu$ ‐special. Proof.. :. Down Lemma for. regressive. of length \geq$\mu$^{+}. \square. mapping f. a. t\in T_{0} which. below. a. Let. \overline{f}:T\rightarrow T. be defined. is. is. a. limit. ordinal}.. regressive and f^{-1\prime\prime}\{t\}. is $\mu$ ‐special. by. \overlin{f}(t)=\le{bginary}{l f(t\mahr{}$\alph);\mathr{i}\mathr{f}$\alph mathr{i}\mathr{s}\mathr{}\mathr{}\mathr{e}\mathr{l}\mathr{}\mathr{}\mathr{g}\mathr{e}\mathr{s}\mathr{}\mathr{l}\mathr{i}\mathr{}\mathr{i}\mathr{}\mathr{o}\mathr{}\mathr{d}\mathr{i}\mathr{n}\mathr{}\mathr{l}\mathr{b}\mathr{e}\mathr{l}\mathr{o}\mathr{w}t_T(+1)\ mathr{}\mathr{}\mathr{e}\mathr{}\mathr{i}\mathr{n}\mathr{i}\mathr{}\mathr{}\mathr{l}\mathr{e}\mathr{l}\mathr{e}\mathr{}\mathr{e}\mathr{n}\mathr{}\mathr{b}\mathr{e}\mathr{l}\mathr{o}\mathr{w};\mathr{i}\mathr{}\mathr{}\mathr{}\mathr{e}\mathr{}\mathr{e}\mathr{i}\mathr{s}\mathr{n}\mathr{o}\mathr{s}\mathr{u}\mathr{c}\mathr{}$\alph. \end{ary}\ight. \overline{f} is regressive.. For. t\in T,. \displaystyle \overline{f}^{-1\prime\prime}\{t\}=\bigcup_{n\in $\omega$} { u\in T. is $\mu$ ‐special since. f^{-1\prime\prime}\{t\}. :. u. is. is $\mu$ ‐special.. an. nth. successor. of. an. element of \square. f^{-1\prime\prime}\{t\} }. (Corollary. 5. 3\rangle.
(12) 45. Proposition $\kappa$\leq cf( $\lambda$) If .. Proof. is. Suppose. 5.4. \mathrm{W}\mathrm{S}_{I }(G_{ $\mu$}^{\downar ow}(< $\kap a$, $\lambda$). Assume that. stationary. that $\mu$, $\kappa$_{f} $\lambda$. in. [\mathcal{M}]^{ $\mu$}.. are. \mathrm{W}\mathrm{S}_{I }(G_{ $\mu$}^{\downar ow}(< $\kap a$, $\lambda$). $\mu$^{< $\mu$}= $\mu$ <$\mu$^{+} < \mathrm{R}\mathrm{C}( $\mu$, < $\kappa,\ \lambda$^{ $\mu$}) does not hold.. cardinals such that. does not hold then. does not hold.. Lemma. By. 4.4, \mathcal{B}. in. (4.37). Let. (5.3). $\tau$=\{\{M_{ $\xi$}. $\xi$\leq $\delta$). :. (a) $\delta$<$\mu$^{+} (b) \{M_{ $\xi$} : $\xi$\leq $\delta$\}. :. ). is. a. continuously. sequence of. elementary submodels of \mathcal{M} of cardinality $\mu$ (c) M_{ $\xi$}\in \mathcal{B} for all successor $\xi$\leq $\delta$ and for all limit $\xi$\leq $\delta$ of cofinality $\mu$, (d) M_{ $\xi$}\in M_{ $\xi$+1}\mathrm{f}\mathrm{o}\mathrm{r} all $\xi$< $\delta$ }. increasing. ,. For t, t' , let. t< $\tau$ t'\Leftrightarrow t. is. initial segment of t'.. an. We show that the tree T. \langle T, < $\tau$ }. =. witnesses the. non. reflection of. non. $\mu$-. specialness1). T\in[ $\eta$< $\kappa$. Claim 5.4.1 All. \vdash. For t\in T with t=. sequence t is. the. \{M_{ $\xi$}. $\mu$ ‐special.. are. :. $\xi$\leq $\delta$\rangle. $\xi$' \mathrm{t}\mathrm{h} component M_{ $\xi$} for $\xi$\leq $\delta$. (5.4). we. denote. \ell_{0}(t)= $\delta$. while the. $\delta$+1. M_{t} denotes the last component M_{ $\delta$} of the .. length. of the. sequence t and. M_{t, $\xi$}. Let. d(t)= $\lambda$\cap M_{t}. and. (5.5). d(T)=\cup\{d(t) : t\in T\}. for T\subseteq T Note that .. on. d(T)\in[ $\lambda$]^{< $\kappa$}. for. T\in[ $\eta$< $\kappa$. .. In. particular, by. a. $\mu$‐special tree. the. assumption. $\lambda$, d(T) for such T is bounded in $\lambda$. We show by induction on $\eta$< $\kappa$ that. (5.6). if. T\in[T]^{< $\kappa$}. holds for all $\eta$< $\kappa$. Suppose (5.6). Clearly. .. this. implies the claim.. .. .. we. .. a successor. Case III: $\eta$ is T \in. [T]^{< $\kappa$}. a. and. ordinal. This. limit ordinal of. otp(d(T)). =. $\eta$. cardinality \geq $\theta$>> $\lambda$.. case. cannot. cofinality \leq $\mu$ .. 1) This tree is not yet the final witness of the it has. then T is. $\eta$<$\mu$^{+} Suppose that T\in[T]^{< $\kappa$} and otp(d(T))= $\eta$ By (5.3), (d), ht(T)<$\mu$^{+} Thus T is $\mu$‐special by Lemma 5.1, (2).. Case II: $\eta$ is. that. otp(d(T))= $\eta$. holds for all $\eta$_{0}< $\eta$.. Case I: have. and. Then. .. we can. negation of. occur. Let. by definition of d(T). .. $\delta$=cf( $\eta$) \leq $\mu$ Suppose. find. .. an. increasing. sequence. \mathrm{R}\mathrm{C}( $\mu$, < $\kap a,\ \lambda$^{ $\mu$}) we are looking for since.
(13) 46. \langle$\xi$_{$\alpha$} T. $\delta$\rangle of ordinals with \displaystyle \sup (\{$\xi$_{ $\alpha$} : $\alpha$ < $\delta$\}) d(T) \subseteq$\xi$_{ $\alpha$}\} for $\alpha$ < $\delta$ Each T_{ $\alpha$} is $\mu$‐special by. : :. <. $\alpha$. induction. .. T=\displaystyle \bigcup_{ $\alpha$< $\delta$}T_{ $\alpha$}. Let T_{ $\alpha$}. .. =. \{t. \in. Hence. hypothesis.. is also $\mu$‐special.. Case IV: $\eta$ is. otp(d(T))= $\eta$ (5.7). \displaystyle \sup(d(T)). =. a. limit ordinal of. Note that. .. T_{0}=T\backslash { t\in T. Since. { t\in T. T_{0} is. $\mu$‐special.. by. the. assumption. > $\mu$. on. $\lambda$. [ $\eta$< $\kappa$ and \displaystyle \sup(d(T))< $\lambda$ Let. Suppose. .. we. have. that T\in. .. t is maximal in T }.. :. t is maximal in T } is. :. cofinality. antichain in T it is. an. to show that. enough. otp(d(T_{0}))<otp(d(T)) then by induction hypothesis T_{0} is special. Hence we that otp(d(T_{0})) may otp(d(T)) (and so \displaystyle \sup(d(T_{0})) \displaystyle \sup(d(T)) ). Let $\nu$=\displaystyle \sup(d(T_{0}))(=\sup(d(T))) and $\delta$=cf( $\nu$) We have $\mu$< $\delta$< $\kappa$ Let \{$\nu$_{ $\beta$} : $\beta$< $\delta$\rangle If. assume. =. =. .. be. a. and. continuously. strictly increasing. .. sequence of ordinals cofinal in. $\nu$>\displaystyle \sup(D_{M_{l}}) holds for all t\in Lim(T_{0}) by Thus, for all t\in Lim(T_{0}) there is f_{t}\in $\lambda$\downarrow $\lambda$\cap M_{t} that. .. Note. T_{0} and (5.3), (d).. the definition of. such that. $\nu$. f_{t}( $\nu$)\displaystyle \geq\sup( $\lambda$\cap M_{t}) by. Lemma 4.3.. t\in Lim(T_{0}) (5.8) Then. \ell_{0}(t). that. Noting. defined at the. and hence. we. h(t)=t \mathrm{r}($\xi$_{0}+1) we. have. h^{-1\prime\prime}\{u\}. \in. u. T_{0}. beginning of the proof. M_{t}=\displaystyle \bigcup_{ $\xi$<l_{0}(t)}M_{t, $\xi$}. where. h:Lim(T_{0})\rightarrow T_{0}. Subclaim 5.4.1.1. \vdash Suppose. have. limit ordinal for. a. ,. $\xi$_{0}=\displaystyle \min\{ $\xi$<\ell_{0}(t) : f_{t}\in M_{t, $\xi$}\}. and h is. is $\mu$ ‐special. Since M_{u} is of. .. is. let. regressive.. for. all. u\in T_{0}.. cardinality. $\mu$ , it is. enough. to show that. f\in $\lambda$\downarrow $\lambda$\cap M_{u} T_{f}=\{t\in h^{-1\prime\prime}\{u\} : f_{t}=t\} f( $\nu$) < $\nu$ there is $\beta$^{*}< $\delta$ such that f(\mathrm{v})<v_{$\beta$^{*} For any t\in T_{f} we have \displaystyle \sup(d(t))\leq f_{t}( $\nu$)= f( $\nu$)\leq \mathrm{v}_{$\beta$^{*} Thus T_{f}\subseteq\{t\in T_{0} : d(t)\subseteq \mathrm{v}_{$\beta$^{*}}\} The suutree of T on the right side of is $\mu$ ‐special for each .. Since. .. .. .. the inclusion is $\mu$‐special. by the induction hypothesis.. Hence. T_{f}. is also $\mu$‐special.. \dashv By Corollary Claim 5.4.2. \vdash Suppose. 5.3 it follows that. $\tau$_{0} is. that. D_{m},. m. < $\mu$. are. \dashv. $\mu$‐special.. open dense subsets of T and t\in T. t'\in T such that t<$\tau$^{t'} and. expansion of the structure \mathcal{M} obtained. M\prec\tilde{\mathcal{M}. (Subclaim 5.4.1.1). T is \leq $\mu$ ‐Baire. Hence it is not $\mu$ ‐special by Lemma 5.1,. to show that there is. Let. ,. ,. be such that. (5.9). t\in M ; and. (5.10). M\in B.. t'\displaystyle \in\bigcap_{m< $\mu$}D_{m}. by adding the. .. Let. unary relations. (Claim. 5.. 4.1\rangle. (1). .. We have. \tilde{\mathcal{M}. be the. D_{m}, m< $\mu$..
(14) 47. There is such M since \mathcal{B} is Let x_{m}, m< $\mu$ be. $\mu$_{0}\}\in M for all $\mu$_{0}< $\mu$. Let \{t_{m} : m< $\mu$\rangle be (5.11). stationary by Lemma. enumeration of M. an. a. 4.4.. Since M satisfies. .. continuously increasing. sequence in. (4.35), \{x_{m}. :. m<. T such that. t_{m}\in M for all m\in$\mu$_{\dot{\text{）} }. (so by the $\mu$_{0}< $\mu$). (5.12). t_{0}=t_{\dot{\text{）} }. (5.13). t_{m+1}\in D_{m}\cap M. reasoning. same. as. above. \{t_{m}. m<$\mu$_{0}\rangle. :. \in. M for all. for all m< $\mu$. (this is possible of M) ;. since. D_{m} is. open dense and. by. the. elementarity. and. (5.14). for all m< $\mu$.. x_{m}\in M_{t_{m+1}}. Let. (5.15). t'=\cup\{t_{m} : m< $\mu$\}^{-}\{M\rangle.. Then t' \in T. by (5.14). and. (5.10).. t. \leq$\tau$^{t'} by (5.12). t'. \in. D_{m} for all. m<. $\mu$. by. \dashv (\mathrm{c} $\iota$ \mathrm{a}\mathrm{i}\mathrm{m}5.4.2). (5.13). Let \mathcal{N}\prec \mathcal{M} be such that. (5.16). $\lambda$\subseteq \mathcal{N}, |\mathcal{N}|=$\lambda$^{< $\mu$} ;. (5.17). [\mathcal{N}\mathrm{J}^{< $\mu$}\subseteq \mathcal{N}.. Let. T_{0}. 5.4.2 also. =. T\cap \mathcal{N}. apply. to. Then. .. |T_{0}|. T_{0} Thus T_{0} .. \leq $\lambda$^{ $\mu$} and the proofs of Claim 5.4.1 and Claim. witnesses that. \mathrm{W}\mathrm{S}_{I }(G_{ $\mu$}^{\downar ow}(< $\kap a$, $\lambda$). does not hold. \square. 6. Reflection. properties related. of Galvins To. to. (Proposition 5.4). generalizations. Conjecture. interpolate the implication from \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}\mathrm{h}\mathrm{r} ( $\mu$, < $\kap a$, $\lambda$). like to prove in this section,. we. to. \mathrm{R}\mathrm{C}( $\mu$, < $\kappa$, $\lambda$). we. would. introduce yet another reflection property which. generalization of Galvins Conjecture. partial ordering \langle P, <P } a subordering P' \langle P', <P^{t} ) of P with P'\underline{\subseteq}P and <P' <P\cap(P')^{2} is said to be a chain if <P' linearly orders P'.. stands in connection with For. a. P. a. =. =. =.
(15) 48. \mathrm{G}\mathrm{C}( $\mu$, < $\kappa$, $\lambda$) :. partial ordering P of cardinality $\lambda$ chains, then there is a subordering P'. For any. \leq $\mu$‐many. that P' is not. Galvins. Conjecture ([15]). \mathrm{G}\mathrm{C}(\aleph_{0}, <\aleph_{2}, $\lambda$). GC:. be formulated. the. of Galvins. consistency. Conjecture. Suppose that. 6.1. $\mu$, $\kappa$, $\lambda$. are. cardinals such that. is. a. long. T and. t_{0}\triangleleft_{T}t_{1}. .. \mathrm{l}\mathrm{e}\mathrm{t}\triangleleft $\tau$ be the binary relation. \Leftrightar ow. t_{0} and t_{1}. are. incomparable. tÓ. and. tí. where. respectively. $\mu$^{+}< $\kappa$\leq $\lambda$.. .. (Todorčevič [16]). (1): Suppose that \mathrm{R}\mathrm{C}( $\mu$, < $\kappa$, $\lambda$) T= \{T, <T\} be a tree of size $\lambda$ witnessing this. Let \triangleleft be. (6.1). such. as:. Proof. on. $\kappa$. holds for all cardinal $\lambda$.. (2) \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}\mathrm{h}\mathrm{r} ( $\mu$, < $\kap a$, $\lambda$) implies \mathrm{G}\mathrm{C}( $\mu$, < $\kappa$, $\lambda$). ordering. union of. \leq $\mu$‐many chains.. (1) \mathrm{G}\mathrm{C}( $\mu$, < $\kappa$, $\lambda$) implies \mathrm{R}\mathrm{C}( $\mu$, < $\kappa$, $\lambda$). let. a. of P of size <. question.. open. Proposition. can. Conjecture,. Unlike Rados. standing. union of. a. if P is not. ,. are. T defined. on. does not hold and. arbitrary linear. an. by. with respect to <_{T} and. tÓ. \triangleleft. tí. minimal elements below t_{0} and t_{1}. with respect to < $\tau$ such that. tÓ. and. tí. are. incomparable It is easy to have. that,. (6.2) Thus. X is. \langle T,. that \triangleleft $\tau$ is. see. for any a. Then,. p\mathcal{E}_{P}q. which is on. p and q. \Leftrightar ow. X is non. Thus. a. a. on. \Leftrightar ow. to. we. X is. an. T. .. By. antichain in. \mathrm{G}\mathrm{C}( $\mu$, < $\kappa$, $\lambda$). the definition. \mathrm{o}\mathrm{f}\triangleleft $\tau$. ,. we. are. \{T,. < $\tau$ ).. .. \{P, <P\} be a partial \mathrm{G}\mathrm{C} ( $\mu$, < $\kappa$, $\lambda$) Let \mathcal{E}_{P} be the. does not hold and let. counterexample. P defined. for any X\subseteq P ,. (6.4). \triangleleft $\tau$ }. \mathrm{G}\mathrm{C}( $\mu$, < $\kappa$, $\lambda$). that. ordering of size $\lambda$ adjacency relation. (6.3). \langle T,. counterexample. a. (2): Suppose. partial ordering. X\subseteq T,. chain in. \triangleleft $\tau$ ) is. a. to. .. by incomparable. with respect to <P.. have. chain in P. (with respect. to <P ). \Leftrightar ow. elements of X. are. pairwise. adjacent.. \{P, \mathcal{E}_{P}\}. is. a. counterexample. to. \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}\mathrm{h}\mathrm{r} ( $\mu$, < $\kap a$, $\lambda$). .. \square. (proposition 6.1).
(16) 49. A. 7 We. proof of. can now. obtain. put together the propositions. proof. a. Theorem 1.1 and we. some. proved. in the. applications previous sections. to. of Theorem 1.1.. Proof of Theorem 1.1:. Suppose. that \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}\mathrm{o}1 ( $\mu$, < $\kappa$, $\lambda$) does not hold.. Let. $\lambda$^{*}\leq $\lambda$ be such that. (7.1). \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}\mathrm{o} | ( $\mu$,. $\lambda$_{0} ) holds for all $\lambda$_{0}<$\lambda$^{*} but \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}\mathrm{o}1}( $\mu$, < $\kap a,\ \lambda$^{*}) does. < $\kappa$ ). not.. By (the proof of) Proposition 3.2) \mathrm{F}\mathrm{R}\mathrm{P}( $\mu$, < $\kappa,\ \lambda$^{*}) does not hold. By Proposi‐ \mathrm{C}\mathrm{C}^{\downar ow} ( $\mu$, < $\kappa,\ \lambda$^{*}) does not hold. Uy Proposition 4.2) this. tion 4.1 it follows that is. equivalent. to the assertion that. \mathrm{W}\mathrm{S}_{I }(G_{ $\mu$}^{\downar ow}(< $\kap a,\ \lambda$^{*}). does not hold.. Proposi‐ implies Thus, by Proposition 6.1) \mathrm{R}\mathrm{C}( $\mu$, < $\kappa$, ($\lambda$^{*})^{ $\mu$}) does not hold. Since it follows by Lemma 2.2, \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}\mathrm{h}\mathrm{r} ( $\mu$, < $\kap a$, ($\lambda$^{*})^{ $\mu$}) ($\lambda$^{*})^{ $\mu$}\leq$\lambda$^{ $\mu$}. tion 5.4. that. now. does not hold.. ,. that. \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}\mathrm{h}\mathrm{r} ( $\mu$, < $\kap a,\ \lambda$^{ $\mu$}). does not hold.. A stationary subset S of. stationary for. Lemma 7.1 Let $\mu$, $\lambda$ be. non‐reflecting. $\xi$\in S). (b). but. =. non‐reflecting if S\cap $\delta$. is not. E_{$\mu$}^{$\lambda$}. with. $\mu$^{+}. Then there is. .. co1 ( G I X ) \leq $\mu$. Suppo \mathcal{S}e that graph G \langle $\lambda$, \mathcal{E}\rangle. < a. $\lambda$. .. =. for all. X \in. [ $\lambda$]^{< $\lambda$}. In. .. there is. a. such that. particular. does not hold.. Without loss of be such. (Theorem 1.1). .. regular cardinals. stationary S \subseteq. $\mu$^{+} (a) co1 (G) \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}\mathrm{o} |( $\mu$, < $\lambda$, $\lambda$) Proof.. cardinal $\lambda$ is said to be. a. $\delta$\in Lim( $\lambda$). any. 0. that,. generality, $\xi$\in S,. we. may. assume. that S\underline{\subseteq}. Lim( $\lambda$). .. Let. \{c_{ $\xi$}. :. for all. (7.2). c_{ $\xi$}\subseteq $\xi$\backslash Lim( $\xi$). (7.3). otp(c_{ $\xi$})= $\mu$.. and c_{ $\xi$} is cofinal in. $\xi$ ;. Let. (7.4). \mathcal{E}=\{( $\alpha$, $\beta$), \langle $\beta$, $\alpha$) : $\alpha$\in S, $\beta$\in c_{ $\alpha$}\}.. Since $\lambda$ is. regular,. the. Claim 7.1.1 For any. \vdash on. Let. C\subseteq $\eta$. $\eta$ such that. is also. an. be. a. following. Claim. $\eta$\in Lim( $\lambda$). ,. implies (b).. co1 (G\mathrm{r} $\eta$)\leq $\mu$.. club subset of $\eta$ such that C\cap S=\emptyset. \mathrm{L}\mathrm{e}\mathrm{t}\triangleleft \mathrm{b}\mathrm{e}. \triangleleft\cap C^{2}=\in\cap c^{2} and, for. open interval between. initial segment of. ( $\alpha$, $\beta$). $\alpha$. and. any $\alpha$,. $\beta$. $\beta$\in C with. well‐ordering. ( $\alpha$, $\beta$)\cap C=\emptyset^{2)}; ( $\alpha$ $\beta$ ) ). with respect \mathrm{t}\mathrm{o}\triangleleft ; and. S\cap( $\alpha$, $\beta$). is. an. with respect \mathrm{t}\mathrm{o}\triangleleft.. \dashv. Then \triangleleft witnesses that co1 (G\mathrm{r} $\eta$)\leq $\mu$.. By the definition of \mathcal{E}(7.4) it is implies (a) and finishes the proof. ,. 2) We denote here with. a. ( $\alpha$, $\beta$). clear that col (G)\leq$\mu$^{+}. the open interval. .. So the. \{ $\xi$< $\eta$ : $\alpha$< $\xi$< $\beta$\}.. (Claim 7.1.1). following Claim.
(17) 50. Claim 7.1.2 co1 (G)\geq$\mu$^{+}.. \vdash Suppose that \triangleleft\mathrm{i}\mathrm{s} an arbitrary well‐ordering of G of order type $\lambda$ (see Lemma 2.1, (1)). By the stationarity of S there is $\xi$^{*} \in S such that $\xi$^{*} is an initial segment with respect to \triangleleft. well‐ordering. of G. But then. .. confirming. \mathcal{E}_$\xi$^{*} $\xi$^{*}. c_{$\xi$^{*} and hence that co1 (G)\leq $\mu$. =. |\mathcal{E}_{$\xi$^{*}^{$\xi$^{*}|. =. $\mu$. .. Thus there is. \dashv. no. (Claim 7.1.2). 0_{(\mathrm{L}\mathrm{e}\mathrm{m}\mathrm{m}\mathrm{a}7.1)}. The. following Theorem. covers some. of the instances of the results in. [11].. If $\mu$ and $\lambda$ are cardinals such that $\mu$^{< $\mu$} $\mu$, $\mu$^{+} < $\lambda$ < $\mu$^{+ $\omega$} and there is a non reflecting stationary set S\subseteq E_{ $\mu$}^{ $\lambda$} then \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}\mathrm{h}r}( $\mu$, < $\lambda,\ \lambda$^{ $\mu$}) does not hold. That is, there is a graph G=\{G, \mathcal{E} ) of cardinality $\lambda$^{ $\mu$} such that \mathrm{c}\mathrm{h}\mathrm{r}(G) > $\mu$ but \mathrm{c}\mathrm{h}\mathrm{r}(G\mathrm{r}X)\leq $\mu$ for all X\in[G]^{< $\lambda$}.. Theorem 7.2. Proof.. By. =. Lemma 7. 1\mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}\mathrm{o} | ( $\mu$, < $\lambda$ ) $\lambda$ ) does not hold.. \mathrm{R}\mathrm{E}\mathrm{F}\mathrm{L}_{\mathrm{c}l,\mathrm{r} ( $\mu$, < $\lambda,\ \lambda$^{ $\mu$}). Hence, by Theorem 1.1,. does not hold.. \square. (Theorem 7.2). References [1]. Akihiro Kanamori, The. [2] Philipp Doebler, forcing. [3]. are. Higher Infinite, Springer‐Verlag (1994/2003).. Rados. Conjecture implies. semiproper, Journal of. Paul Erdós and Andrss. that all. stationary set preserving Mathematical Logic Vol.13, 1 (2013).. Hajnal, On chromatic number of graphs Hungarica, 17 (1‐2), (1966), 61‐99.. and set‐. systems, Acta Mathematica. [4]. Sakaé Fuchino,. Juhasz, Lajos Soukup, Zoltán Szentmiklóssy and. István. Toshimichi Usuba,. ity. and. Fodor‐type Reflection Principle and reflection of metrizabil‐ meta‐Lindelöfness, Topology and its Applications Vol.157, 8 (2010),. 1415‐1429.. [5]. Sakaé Fuchino, about. [6]. Lajos Soukup, Hiroshi Sakai and Toshimichi Usuba, More Fodor‐type Reflection Principle, submitted.. Sakaé Fuchino and Assaf Rinot,. Openly generated Boolean algebras and the. Fodor‐type Reflection Principle, Fundamenta. Mathematicae 212,. (2011),. 261‐. 283.. [7]. Sakaé Fuchino and Hiroshi list‐chromatic number of. [8]. Sakaé Fuchino, Hiroshi Rados. Conjecture. and. Sakai, On reflection and non‐reflection of countable graphs, Kôkyûroku, No.1790, (2012)) 31‐44.. Sakai, Victor Torres‐Perez and Toshimichi Usuba, the Fodor‐type Reflection Principle, in preparation..
(18) 51. [9]. Sakaé Fuchino, André Ottenbreit Maschio flection of properties of uncountable. [10]. A.. Hajnal and I. Juhász, On spaces in able, Bulletin de lacadémie polonaise. astr. et. [11]. Rodrigues and Hiroshi Sakai, characteristics, in preparation.. phys., Vol.24, No.9, (1976),. which every small des. subspace. Hungarica 139(4) (2013),. 728‐731.. Stevo. [13]. Stevo. [14]. Stevo Todorčevič, Real functions. Todorčevič, Stationary sets, Beograd 43 (1981), 249‐262.. trees and. continuums, Publ. Inst. Math.. Todorčevič, On a conjecture of Rado, Journal Society Vol.s2‐27, (1) (1983), 1‐8.. partially ordered set,. graphs, Acta. 363‐371.. [12]. a. is metriz‐. sciences, Serie des sciences math,. Saharon Shelah, On incompactness for chromatic number of Math. Re‐. on. London Mathematical. family of all well‐ordered subsets of Symbolic Logic, Vol.48, No.1 (1983),. the. the Journal of. 91‐96.. [15]. Stevo. Todorčevič, Conjectures of Rado. in: Finite and Infinite Sets in. Acad. Publ.. [16]. (1993). and. Chang and Cardinal Arithmetic, Combinatorics (N. W. Sauer et al., eds), Kluwer. 385‐398.. Stevo Todorcevic, Combinatorial dichotomies in set. Symbolic Logic, Vol.17,. No.1. (2011),. 1‐72.. theory,. the Bulletin of.
(19)
関連したドキュメント
In Section 3, we show that the clique- width is unbounded in any superfactorial class of graphs, and in Section 4, we prove that the clique-width is bounded in any hereditary
Includes some proper curves, contrary to the quasi-Belyi type result.. Sketch of
John Baez, University of California, Riverside: [email protected] Michael Barr, McGill University: [email protected] Lawrence Breen, Universit´ e de Paris
(The definition of this invariant given in [13] is somewhat different from the one we use, which comes from [23], but the two definitions can be readily shown to agree.) Furuta and
In particular this implies a shorter and much more transparent proof of the combinatorial part of the Mullineux conjecture with additional insights (Section 4). We also note that
Our bound does not prove that every Cayley graph is a ˇ Cerný Cayley graph, but it does work for certain Cayley graphs of cyclic groups, dihedral groups, sym- metric groups,
Theorem (B-H-V (2001), Abouzaid (2006)) A classification of defective Lucas numbers is obtained:.. Finitely many
Platonov conjectured, conversely, that finitely generated linear groups which are super- rigid must be of “arithmetic type.” We construct counterexamples to Platonov’s
