Poset Ramsey Number R(P, Qn). I. Complete Multipartite Posets

A poset $(P′,≤_{P′})$ contains a copy of some other poset $(P,≤_P)$ if there is an injection $f:P′→P$ where for every $X,Y∈P, X≤_PY$ if and only if $f(X)≤_{P′}f(Y)$. For any posets $P$ and $Q$, the poset Ramsey number $R(P, Q)$ is the smallest integer $N$ such that any blue/red coloring of a Boolean lattice of dimension $N$ contains either a copy of $P$ with all elements blue or a copy of $Q$ with all elements red. A complete ℓ-partite poset $K_{t1,…,tℓ}$ is a poset on $∑^ℓ_{i=1}t_i$ elements, which are partitioned into $ℓ$ pairwise disjoint sets $A^i$ with $|A^i|=t_i, 1≤i≤ℓ$, such that for any two $X∈A^i$ and $Y∈A^j$, $X<Y$ if and only if $i<j$. In this paper we show that $R(K_{t1,…,tℓ, }Q_n)≤n+\frac{(2+on(1))ℓn}{logn}$

On ordered Ramsey numbers of matchings versus triangles

For graphs $G^<$ and $H^<$ with linearly ordered vertex sets, the \ordered Ramsey number $r_<(G^<,H^<)$ is the smallest positive integer $N$ such that any red-blue coloring of the edges of the complete ordered graph $K^<_N$ on $N$ vertices contains either a blue copy of $G^<$ or a red copy of $H^<$. Motivated by a problem of Conlon, Fox, Lee, and Sudakov (2017), we study the numbers $r_<(M^<,K^<_3)$ where $M^<$ is an ordered matching on $n$ vertices. We prove that almost all $n$-vertex ordered matchings $M^<$ with interval chromatic number 2 satisfy $r_<(M^<,K^<_3) \in \Omega((n/\log n)^{5/4})$ and $r_<(M^<,K^<_3) \in O(n^{7/4})$, improving a recent result by Rohatgi (2019). We also show that there are $n$-vertex ordered matchings $M^<$ with interval chromatic number at least 3 satisfying $r_<(M^<,K^<_3) \in \Omega((n/\log n)^{4/3})$, which asymptotically matches the best known lower bound on these off-diagonal ordered Ramsey numbers for general $n$-vertex ordered matchings.Comment: 16 pages, 2 figures; extended abstract to appear at EuroComb 202

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

Generalized cohesiveness

We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey's Theorem. An infinite set $A$ of natural numbers is $n$--cohesive (respectively, $n$--r--cohesive) if $A$ is almost homogeneous for every computably enumerable (respectively, computable) $2$--coloring of the $n$--element sets of natural numbers. (Thus the $1$--cohesive and $1$--r--cohesive sets coincide with the cohesive and r--cohesive sets, respectively.) We consider the degrees of unsolvability and arithmetical definability levels of $n$--cohesive and $n$--r--cohesive sets. For example, we show that for all $n \ge 2$, there exists a $\Delta^0_{n+1}$ $n$--cohesive set. We improve this result for $n = 2$ by showing that there is a $\Pi^0_2$ $2$--cohesive set. We show that the $n$--cohesive and $n$--r--cohesive degrees together form a linear, non--collapsing hierarchy of degrees for $n \geq 2$. In addition, for $n \geq 2$ we characterize the jumps of $n$--cohesive degrees as exactly the degrees {\bf \geq \jump{0}{(n+1)}} and show that each $n$--r--cohesive degree has jump {\bf > \jump{0}{(n)}}

Algorithms and almost tight results for 3-colorability of small diameter graphs.

The 3-coloring problem is well known to be NP-complete. It is also well known that it remains NP-complete when the input is restricted to graphs with diameter 4. Moreover, assuming the Exponential Time Hypothesis (ETH), 3-coloring cannot be solved in time 2o(n) on graphs with n vertices and diameter at most 4. In spite of extensive studies of the 3-coloring problem with respect to several basic parameters, the complexity status of this problem on graphs with small diameter, i.e. with diameter at most 2, or at most 3, has been an open problem. In this paper we investigate graphs with small diameter. For graphs with diameter at most 2, we provide the first subexponential algorithm for 3-coloring, with complexity 2O(nlogn√). Furthermore we extend the notion of an articulation vertex to that of an articulation neighborhood, and we provide a polynomial algorithm for 3-coloring on graphs with diameter 2 that have at least one articulation neighborhood. For graphs with diameter at most 3, we establish the complexity of 3-coloring by proving for every ε∈[0,1) that 3-coloring is NP-complete on triangle-free graphs of diameter 3 and radius 2 with n vertices and minimum degree δ=Θ(nε). Moreover, assuming ETH, we use three different amplification techniques of our hardness results, in order to obtain for every ε∈[0,1) subexponential asymptotic lower bounds for the complexity of 3-coloring on triangle-free graphs with diameter 3 and minimum degree δ=Θ(nε). Finally, we provide a 3-coloring algorithm with running time 2O(min{δΔ, nδlogδ}) for arbitrary graphs with diameter 3, where n is the number of vertices and δ (resp. Δ) is the minimum (resp. maximum) degree of the input graph. To the best of our knowledge, this is the first subexponential algorithm for graphs with δ=ω(1) and for graphs with δ=O(1) and Δ=o(n). Due to the above lower bounds of the complexity of 3-coloring, the running time of this algorithm is asymptotically almost tight when the minimum degree of the input graph is δ=Θ(nε), where ε∈[12,1), as its time complexity is 2O(nδlogδ)=2O(n1−εlogn) and the corresponding lower bound states that there is no 2o(n1−ε)-time algorithm