Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse

We work with symmetric extensions based on L\'{e}vy Collapse and extend a few results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her P.h.d. thesis. We also observe that if $V$ is a model of ZFC, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-distributive and $\mathcal{F}$ is $\kappa$-complete. Further we observe that if $V$ is a model of ZF + $DC_{\kappa}$, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-strategically closed and $\mathcal{F}$ is $\kappa$-complete.Comment: Revised versio

Chromatic number of the product of graphs, graph homomorphisms, Antichains and cofinal subsets of posets without AC

We have observations concerning the set theoretic strength of the following combinatorial statements without the axiom of choice. 1. If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. 2. If in a partially ordered set, all chains are finite and all antichains have size $\aleph_{\alpha}$, then the set has size $\aleph_{\alpha}$ for any regular $\aleph_{\alpha}$. 3. CS (Every partially ordered set without a maximal element has two disjoint cofinal subsets). 4. CWF (Every partially ordered set has a cofinal well-founded subset). 5. DT (Dilworth's decomposition theorem for infinite p.o.sets of finite width). 6. If the chromatic number of a graph $G_{1}$ is finite (say $k<\omega$), and the chromatic number of another graph $G_{2}$ is infinite, then the chromatic number of $G_{1}\times G_{2}$ is $k$. 7. For an infinite graph $G=(V_{G}, E_{G})$ and a finite graph $H=(V_{H}, E_{H})$, if every finite subgraph of $G$ has a homomorphism into $H$, then so has $G$. Further we study a few statements restricted to linearly-ordered structures without the axiom of choice.Comment: Revised versio

Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC

In set theory without the Axiom of Choice (AC), we observe new relations of the following statements with weak choice principles. • Plf,c (Every locally finite connected graph has a maximal independent set). • Plc,c (Every locally countable connected graph has a maximal independent set). • CACאα (If in a partially ordered set all antichains are finite and all chains have size אα, then the set has size אα) if אα is regular. • CWF (Every partially ordered set has a cofinal well-founded subset). • If G = (VG, EG) is a connected locally finite chordal graph, then there is an ordering <of VG such that {w < v : {w, v} ∈ EG} is a clique for each v ∈ VG

Combinatorial properties and dependent choice in symmetric extensions based on Lévy collapse

We work with symmetric extensions based on Lévy collapse and extend a few results of Apter, Cody, and Koepke. We prove a conjecture of Dimitriou from her Ph.D. thesis. We also observe that if V is a model of ZFC, then DC<κ can be preserved in the symmetric extension of V in terms of symmetric system ⟨ P, G, F⟩ , if P is κ-distributive and F is κ-complete. Further we observe that if δ< κ and V is a model of ZF+ DCδ, then DCδ can be preserved in the symmetric extension of V in terms of symmetric system ⟨ P, G, F⟩ , if P is (δ+ 1)-strategically closed and F is κ-complete. © 2022, The Author(s)

Partition models, Permutations of infinite sets without fixed points, and weak forms of AC

Abstract. In set theory without the Axiom of Choice (AC), we observe new relations of the following statements with weak choice forms. • There does not exist an infinite Hausdorff space X such that every infinite subset of X contains an infinite compact subset. • If a field has an algebraic closure then it is unique up to isomorphism. • For every set X there is a set A such that there exists a choice function on the collection [A] 2 of two-element subsets of A and satisfying |X| ≤ |2 [A] 2 |. • Van Douwen’s Choice Principle (Every family X = {(Xi, ≤i) : i ∈ I} of linearly ordered sets isomorphic with (Z, ≤) has a choice function, where ≤ is the usual ordering on Z). We also extend the research works of B.B. Bruce [4]. Moreover, we prove that the principle “Any infinite locally finite connected graph has a spanning m-bush for any even integer m ≥ 4” is equivalent to K˝onig’s Lemma in ZF (i.e., the Zermelo–Fraenkel set theory without AC). We also give a new combinatorial proof to show that any infinite locally finite connected graph has a chromatic number if and only if K˝onig’s Lemma holds

Chromatic number of the product of graphs, graph homomorphisms, antichains and cofinal subsets of posets without AC

In set theory without the axiom of choice (AC), we observe new relations of the following statements with weak choice principles. ◦ If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable.If in a partially ordered set, all chains are finite and all antichains have size Nα, then the set has size Nα for any regular Nα. Every partially ordered set without a maximal element has two disjoint cofinal sub sets – CS. Every partially ordered set has a cofinal well-founded subset – CWF. Dilworth’s decomposition theorem for infinite partially ordered sets of finite width – DT. We also study a graph homomorphism problem and a problem due to A. Hajnal without AC. Further, we study a few statements restricted to linearly-ordered structures without AC. © 2021, Commentationes Mathematicae Universitatis Carolinae. All Rights Reserved

Chromatic number of the product of graphs, graph homomorphisms, antichains and cofinal subsets of posets without AC

summary:In set theory without the axiom of choice (AC), we observe new relations of the following statements with weak choice principles. $\circ$ If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. $\circ$ If in a partially ordered set, all chains are finite and all antichains have size $\aleph_{\alpha}$, then the set has size $\aleph_{\alpha}$ for any regular $\aleph_{\alpha}$. $\circ$ Every partially ordered set without a maximal element has two disjoint cofinal sub sets -- CS. $\circ$ Every partially ordered set has a cofinal well-founded subset -- CWF. $\circ$ Dilworth's decomposition theorem for infinite partially ordered sets of finite width -- DT. We also study a graph homomorphism problem and a problem due to A. Hajnal without AC. Further, we study a few statements restricted to linearly-ordered structures without AC