Some remarks on off-diagonal Ramsey numbers for vector spaces over F2\mathbb{F}_{2}

For every positive integer $d$, we show that there must exist an absolute constant $c > 0$ such that the following holds: for any integer $n \geq cd^{7}$ and any red-blue coloring of the one-dimensional subspaces of $\mathbb{F}_{2}^{n}$, there must exist either a $d$-dimensional subspace for which all of its one-dimensional subspaces get colored red or a $2$-dimensional subspace for which all of its one-dimensional subspaces get colored blue. This answers recent questions of Nelson and Nomoto, and confirms that for any even plane binary matroid $N$, the class of $N$-free, claw-free binary matroids is polynomially $\chi$-bounded. Our argument will proceed via a reduction to a well-studied additive combinatorics problem, originally posed by Green: given a set $A \subset \mathbb{F}_{2}^{n}$ with density $\alpha \in [0,1]$, what is the largest subspace that we can find in $A+A$? Our main contribution to the story is a new result for this problem in the regime where $1/\alpha$ is large with respect to $n$, which utilizes ideas from the recent breakthrough paper of Kelley and Meka on sets of integers without three-term arithmetic progressions

A tight linear chromatic bound for (P3∪P2,W4P_3\cup P_2, W_4)-free graphs

For two vertex disjoint graphs $H$ and $F$, we use $H\cup F$ to denote the graph with vertex set $V(H)\cup V(F)$ and edge set $E(H)\cup E(F)$, and use $H+F$ to denote the graph with vertex set $V(H)\cup V(F)$ and edge set $E(H)\cup E(F)\cup\{xy\;|\; x\in V(H), y\in V(F)$$\}$. A $W_4$ is the graph $K_1+C_4$. In this paper, we prove that $\chi(G)\le 2\omega(G)$ if $G$ is a ($P_3\cup P_2, W_4$)-free graph. This bound is tight when $\omega =2$ and $3$, and improves the main result of Wang and Zhang. Also, this bound partially generalizes some results of Prashant {\em et al.}.Comment: arXiv admin note: text overlap with arXiv:2308.05442, arXiv:2307.1194

Coloring_of_some_crown-free_graphs

Let $G$ and $H$ be two vertex disjoint graphs. The {\em union} $G\cup H$ is the graph with $V(G\cup H)=V(G)\cup (H)$ and $E(G\cup H)=E(G)\cup E(H)$. The {\em join} $G+H$ is the graph with $V(G+H)=V(G)+V(H)$ and $E(G+H)=E(G)\cup E(H)\cup\{xy\;|\; x\in V(G), y\in V(H)$$\}$. We use $P_k$ to denote a {\em path} on $k$ vertices, use {\em fork} to denote the graph obtained from $K_{1,3}$ by subdividing an edge once, and use {\em crown} to denote the graph $K_1+K_{1,3}$. In this paper, we show that (\romannumeral 1) $\chi(G)\le\frac{3}{2}(\omega^2(G)-\omega(G))$ if $G$ is (crown, $P_5$)-free, (\romannumeral 2) $\chi(G)\le\frac{1}{2}(\omega^2(G)+\omega(G))$ if $G$ is (crown, fork)-free, and (\romannumeral 3) $\chi(G)\le\frac{1}{2}\omega^2(G)+\frac{3}{2}\omega(G)+1$ if $G$ is (crown, $P_3\cup P_2$)-free.Comment: arXiv admin note: text overlap with arXiv:2302.0680

Hardness of Approximating Bounded-Degree Max 2-CSP and Independent Set on k-Claw-Free Graphs

We consider the question of approximating Max 2-CSP where each variable appears in at most $d$ constraints (but with possibly arbitrarily large alphabet). There is a simple $(\frac{d+1}{2})$-approximation algorithm for the problem. We prove the following results for any sufficiently large $d$: - Assuming the Unique Games Conjecture (UGC), it is NP-hard (under randomized reduction) to approximate this problem to within a factor of $\left(\frac{d}{2} - o(d)\right)$. - It is NP-hard (under randomized reduction) to approximate the problem to within a factor of $\left(\frac{d}{3} - o(d)\right)$. Thanks to a known connection [Dvorak et al., Algorithmica 2023], we establish the following hardness results for approximating Maximum Independent Set on $k$-claw-free graphs: - Assuming the Unique Games Conjecture (UGC), it is NP-hard (under randomized reduction) to approximate this problem to within a factor of $\left(\frac{k}{4} - o(k)\right)$. - It is NP-hard (under randomized reduction) to approximate the problem to within a factor of $\left(\frac{k}{3 + 2\sqrt{2}} - o(k)\right) \geq \left(\frac{k}{5.829} - o(k)\right)$. In comparison, known approximation algorithms achieve $\left(\frac{k}{2} - o(k)\right)$-approximation in polynomial time [Neuwohner, STACS 2021; Thiery and Ward, SODA 2023] and $(\frac{k}{3} + o(k))$-approximation in quasi-polynomial time [Cygan et al., SODA 2013]

Connected k-Partition of k-Connected Graphs and c-Claw-Free Graphs

w_k. In particular for the balanced version, i.e. w? = w? == w_k, this gives a partition with 1/3w_i ? w(T_i) ? 3w_i