New Lower Bound Formulas for Multicolored Ramsey Numbers

Large chromatic number and Ramsey graphs

Let Q(n,c) denote the minimum clique size an n-vertex graph can have if its chromatic number is c. Using Ramsey graphs we give an exact, albeit implicit, formula for the case c is at least (n+3)/2.

Defective Ramsey Numbers in Graph Classes

Given a graph $G$, a $k$-sparse $j$-set is a set of $j$ vertices inducing a subgraph with maximum degree at most $k$. A $k$-dense $i$-set is a set of $i$ vertices that is $k$-sparse in the complement of $G$. As a generalization of Ramsey numbers, the $k$-defective Ramsey number $R_k^{\mathcal{G}}(i,j)$ for the graph class $\mathcal{G}$ is defined as the smallest natural number $n$ such that all graphs on $n$ vertices in the class $\mathcal{G}$ have either a $k$-dense $i$-set or a $k$-sparse $j$-set. In this paper, we examine $R_k^{\mathcal{G}}(i,j)$ where $\mathcal{G}$ represents various graph…

A Ramsey-type theorem for the matching number regarding connected graphs

A major line of research is discovering Ramsey-type theorems, which are results of the following form: given a graph parameter $ρ$, every graph $G$ with sufficiently large $ρ(G)$ contains a `well-structured' induced subgraph $H$ with large $ρ(H)$. The classical Ramsey's theorem deals with the case when the graph parameter under consideration is the number of vertices; there is also a Ramsey-type theorem regarding connected graphs. Given a graph $G$, the matching number and the induced matching number of $G$ is the maximum size of a matching and an induced matching, respectively, of $G$. In thi…

On path-quasar Ramsey numbers

Let $G_1$ and $G_2$ be two given graphs. The Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\overline{G}$ contains a $G_2$. Parsons gave a recursive formula to determine the values of $R(P_n,K_{1,m})$, where $P_n$ is a path on $n$ vertices and $K_{1,m}$ is a star on $m+1$ vertices. In this note, we first give an explicit formula for the path-star Ramsey numbers. Secondly, we study the Ramsey numbers $R(P_n,K_1\vee F_m)$, where $F_m$ is a linear forest on $m$ vertices. We determine the exact values of $R(P_n,K_1…

Refined Turán numbers and Ramsey numbers for the loose 3-uniform path of length three

Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$-color Ramsey number for $P$ is $R(P;r)=r+6$ for $r\le 7$. The proof of this result relies on a careful analysis of the Turán numbers for $P$. In this paper, we refine this analysis further and compute, for all $n$, the third and fourth order Turán numbers for $P$. With the help of the former, we confirm the formula $R(P;r)=r+6$ for $r\in\{8,9\}$.

One more Turán number and Ramsey number for the loose 3-uniform path of length three

Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$-color Ramsey number for $P$ is $R(P;r)=r+6$ for $r\le 9$. The proof of this result relies on a careful analysis of the Turán numbers for $P$. In this paper, we refine this analysis further and compute the fifth order Turán number for $P$, for all $n$. Using this number for $n=16$, we confirm the formula $R(P;10)=16$.

Turán's problem and Ramsey numbers for trees

Let $T_n^1=(V,E_1)$ and $T_n^2=(V,E_2)$ be the trees on $n$ vertices with $V=\{v_0,v_1,\ldots,v_{n-1}\}$, $E_1=\{v_0v_1,\ldots,v_0v_{n-3},v_{n-4}v_{n-2},v_{n-3}v_{n-1}\}$, and $E_2=\{v_0v_1,\ldots,$ $v_0v_{n-3},v_{n-3}v_{n-2}, v_{n-3}v_{n-1}\}$. In this paper, for $p\ge n\ge 5$ we obtain explicit formulas for $\ex(p;T_n^1)$ and $\ex(p;T_n^2)$, where $\ex(p;L)$ denotes the maximal number of edges in a graph of order $p$ not containing $L$ as a subgraph. Let $r(G\sb 1, G\sb 2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. In this paper we also obtain some explicit formulas for $r(T_m,…

The pigenhole principle and multicolor Ramsey numbers

For integers $k,r\geq 2$, the diagonal Ramsey number $R_r(k)$ is the minimum $N\in\mathbb{N}$ such that every $r$-coloring of the edges of a complete graph on $N$ vertices yields on a monochromatic subgraph on $k$ vertices. Here we make a careful effort of extracting explicit upper bounds for $R_r(k)$ from the pigeonhole principle alone. Our main term improves on previously documented explicit bounds for $r\geq 3$, and we also consider an often ignored secondary term, which allows us to subtract a uniformly bounded below positive proportion of the main term. Asymptotically, we give a self-cont…

Ramsey numbers of ordered graphs

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by two colors contains a monochromatic copy of $\mathcal{G}$. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which $\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that ordered Ramsey numbers of the same…

Ramsey-type results for semi-algebraic relations

A k-ary semi-algebraic relation E on R^d is a subset of R^{kd}, the set of k-tuples of points in R^d, which is determined by a finite number of polynomial equations and inequalities in kd real variables. The description complexity of such a relation is at most t if the number of polynomials and their degrees are all bounded by t. A subset A of R^d is called homogeneous if all or none of the k-tuples from A satisfy E. A large number of geometric Ramsey-type problems and results can be formulated as questions about finding large homogeneous subsets of sets in R^d equipped with semi-algebraic rel…

Ramsey numbers of cliques versus monotone paths

One formulation of the Erdos-Szekeres monotone subsequence theorem states that for any red/blue coloring of the edge set of the complete graph on $\{1, 2, \ldots, N\}$, there exists a monochromatic red $s$-clique or a monochromatic blue increasing path $P_n$ with $n$ vertices, provided $N >(s-1)(n-1)$. %We had previously shown that a suitable generalization of this problem to quadruple systems is essentially equivalent to classical diagonal hypergraph Ramsey numbers. Here, we prove a similar statement as above in the off-diagonal case for triple systems, with the quasipolynomial bound $N>2^{c(…

Multicolor Ramsey numbers and restricted Turán numbers for the loose 3-uniform path of length three

Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$-colored Ramsey number for $P$ is $R(P;r)=r+6$ for $r=2,3$, and that $R(P;r)\le 3r$ for all $r\ge3$. The latter result follows by a standard application of the Turán number $ex_3(n;P)$, which was determined to be $\binom{n-1}2$ in our previous work. We have also shown that the full star is the only extremal 3-graph for $P$. In this paper, we perform a subtle analysis of the Turán numbers for $P$ under some additional restrictions. Most impo…

On Some Three-Color Ramsey Numbers for Paths

For graphs $G_1, G_2, G_3$, the three-color Ramsey number $R(G_1,$ $G_2, G_3)$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph of order $n$ with 3 colors, then it contains a monochromatic copy of $G_i$ in color $i$, for some $1 \leq i \leq 3$. First, we prove that the conjectured equality $R_3(C_{2n},C_{2n},C_{2n})=4n$, if true, implies that $R_3(P_{2n+1},P_{2n+1},P_{2n+1})=4n+1$ for all $n \ge 3$. We also obtain two new exact values $R(P_8,P_8,P_8)=14$ and $R(P_9,P_9,P_9)=17$, furthermore we do so without help of computer algorithms. Our results a…

Two problems in graph Ramsey theory

We study two problems in graph Ramsey theory. In the early 1970's, Erdős and O'Neil considered a generalization of Ramsey numbers. Given integers $n,k,s$ and $t$ with $n \ge k \ge s,t \ge 2$, they asked for the least integer $N=f_k(n,s,t)$ such that in any red-blue coloring of the $k$-subsets of $\{1, 2,\ldots, N\}$, there is a set of size $n$ such that either each of its $s$-subsets is contained in some red $k$-subset, or each of its $t$-subsets is contained in some blue $k$-subset. Erdős and O'Neil found an exact formula for $f_k(n,s,t)$ when $k\ge s+t-1$. In the arguably more interesting ca…

Polynomial to exponential transition in Ramsey theory

Given $s \ge k\ge 3$, let $h^{(k)}(s)$ be the minimum $t$ such that there exist arbitrarily large $k$-uniform hypergraphs $H$ whose independence number is at most polylogarithmic in the number of vertices and in which every $s$ vertices span at most $t$ edges. Erd\H os and Hajnal conjectured (1972) that $h^{(k)}(s)$ can be calculated precisely using a recursive formula and Erd\H os offered \$500 for a proof of this. For $k=3$ this has been settled for many values of $s$ including powers of three but it was not known for any $k\geq 4$ and $s\geq k+2$. Here we settle the conjecture for all $s \g…

顶点极小Ramsey数：精确值和极值结构

Vertex-minor Ramsey numbers: exact values and extremal structure

本文确定了顶点极小Ramsey数Rvm(4)=11，并通过局部补运算对10和11顶点的图进行穷举分类，揭示了极值结构及方法。

Phase transitions in the Ramsey-Turán theory

Let $f(n)$ be a function and $L$ be a graph. Denote by $RT(n,L,f(n))$ the maximum number of edges of an $L$-free graph on $n$ vertices with independence number less than $f(n)$. Erd\H os and Sós asked if $RT\left(n, K_5, c\sqrt{n}\right) = o(n^2)$ for some constant $c$. We answer this question by proving the stronger $RT\left(n, K_5, o\left(\sqrt{n\log n}\right)\right) = o(n^2)$. It is known that $RT \left(n, K_5, c \sqrt{n\log n} \right) = n^2/4+o(n^2)$ for $c>1$, so one can say that $K_5$ has a Ramsey-Turán phase transition at $c\sqrt{n\log n}$. We extend this result to several other $K_s$'s…

The chromatic gap and its extremes

The {\em chromatic gap} is the difference between the chromatic number and the clique number of a graph. Here we investigate $\gap(n)$, the maximum chromatic gap over graphs on $n$ vertices. Can the extremal graphs be explored? While computational problems related to the chromatic gap are hopeless, an interplay between Ramsey theory and matching theory leads to a simple and (almost) exact formula for $\gap(n)$ in terms of Ramsey numbers.

Ramsey numbers for trees II

Let $r(G_1, G_2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. For $n_1\ge n_2\ge 1$ let $S(n_1,n_2)$ be the double star given by $V(S(n_1,n_2))=\{v_0,v_1,\ldots,v_{n_1},w_0,w_1,\ldots,w_{n_2}\}$ and $E(S(n_1,n_2))=\{v_0v_1,\ldots,v_0v_{n_1},v_0w_0,w_0w_1,\ldots,w_0w_{n_2}\}$. In this paper we determine $r(K_{1,m-1},$ $S(n_1,n_2))$ under certain conditions. For $n\ge 6$ let $T_n^3=S(n-5,3)$, $T_n^{''}=(V,E_2)$ and $T_n^{'''} =(V,E_3)$, where $V=\{v_0,v_1,\ldots,v_{n-1}\}$, $E_2=\{v_0v_1,\ldots,v_0v_{n-4},v_1v_{n-3},v_1v_{n-2},$ $v_2v_{n-1}\}$ and $E_3=\{v_0v_1,\ldots,v_0v_{n-4},v_1v…

Fraisse Structures with SDAP+, Part I: Indivisibility

This is Part I of a two-part series regarding Ramsey properties of Fraisse structures satisfying a property called SDAP+, which strengthens the Disjoint Amalgamation Property. We prove that every Fraisse structure in a finite relational language with relation symbols of any finite arity satisfying this property is indivisible. Novelties include a new formulation of coding trees in terms of 1-types over initial segments of the Fraisse structure, and a direct proof of indivisibility which uses the method of forcing to conduct unbounded searches for finite sets. In Part II, we prove that every Fr…

关于SAT求解器与Ramsey型数

On SAT Solvers and Ramsey-type Numbers

本文通过并行化SAT求解器研究Ramsey型数，得到$R_3(L)$的新界$21 \leq R_3(L) \leq 2593$，并首次确定$R_4(VS)=58$及$R_5(VS)>180$、$R_6(VS)>333$、$R_3(VC)>521$等新结果。

关于涉及奇圈的三个拉姆齐数的研究

A study of two Ramsey numbers involving odd cycles

本文研究涉及奇圈的拉姆齐数，改进了书图与圈拉姆齐数的界，并确定了完全二分图与奇圈拉姆齐数的精确值，证明其具有拉姆齐好性。

关于小有序和循环Ramsey数的一些结果

Some results on small ordered and cyclic Ramsey numbers

本文研究了小有序和循环Ramsey数，通过Kissat求解器确定了多种图类的Ramsey数，并引入了循环Ramsey数概念，探讨了强化学习在Ramsey数下界计算中的应用。

Ramsey numbers of path-matchings, covering designs and 1-cores

A path-matching of order $p$ is a vertex disjoint union of nontrivial paths spanning $p$ vertices. Burr and Roberts, and Faudree and Schelp determined the 2-color Ramsey number of path-matchings. In this paper we study the multicolor Ramsey number of path-matchings. Given positive integers $r, p_1, \dots, p_r$, define $R^{PM}(p_1, \dots, p_r)$ to be the smallest integer $n$ such that in any $r$-coloring of the edges of $K_n$ there exists a path-matching of color $i$ and order at least $p_i$ for some $i\in [r]$. Our main result is that for $r\geq 2$ and $p_1\geq \dots\geq p_r\geq 2$, if $p_1\ge…

Strong Ramsey Games in Unbounded Time

For two graphs $B$ and $H$ the strong Ramsey game $\mathcal{R}(B,H)$ on the board $B$ and with target $H$ is played as follows. Two players alternately claim edges of $B$. The first player to build a copy of $H$ wins. If none of the players win, the game is declared a draw. A notorious open question of Beck asks whether the first player has a winning strategy in $\mathcal{R}(K_n,K_k)$ in bounded time as $n\rightarrow\infty$. Surprisingly, in a recent paper Hefetz et al. constructed a $5$-uniform hypergraph $\mathcal{H}$ for which they proved that the first player does not have a winning strate…

小反射和二面体拉姆齐数的计算

Computation of small reflective and dihedral Ramsey numbers

研究小反射和二面体拉姆齐数的计算问题，应用基于SAT的方法，利用Kissat SAT求解器，针对特定图类得到小反射和二面体拉姆齐数的精确值与下界，并得出一般结果和提出猜想。

$k$次幂Paley有向图的传递子竞赛图与Ramsey数的改进下界

Transitive subtournaments of $k$-th power Paley digraphs and improved lower bounds for Ramsey numbers

本文研究$k$次幂Paley有向图中的传递子竞赛图计数，利用有限域超几何函数与Jacobi和得到三、四阶子竞赛图数量公式，并据此改进多色有向Ramsey数下界。

A construction for weak Schur partitions

In 1952, J.H.Braun claimed to have established a formula giving a lower bound for certain partitions of sets of integers into weakly sum-free classes. However, no proof or supporting construction was published at that time. In today's terminology, that claim was equivalent to giving a formulaic lower bound for the weak Schur number $WS(s)$. $WS(s)$ is the maximum number such that there exists a weak Schur partition of the integers from 1 to $WS(s)$, into $s$ subsets. In a weak Schur partition of a set of integers, there can be no three distinct members $a$, $b$ and $c$ in any subset, such that…

模曲线模空间中的符号反转自映射

Sign-reversing involutions in moduli spaces of curves

本文利用符号反转自映射解决模曲线模空间几何中的两个计算问题，给出 genus 零多色空间上 ψ 类交积的组合公式，并研究图形模空间的热带欧拉特征。

精确顶点互质拉姆齐数的素数证书

Prime Certificates for Exact Vertex-Coprime Ramsey Numbers

通过素数团和素数箱划分，给出了顶点着色互质拉姆齐数的精确公式，并推广到边着色情形。

排他性原理、Ramsey理论与$n$-循环PR盒

Exclusivity principle, Ramsey theory, and $n$-cycle PR boxes

本文将多副本排他性原理违反问题转化为边着色排他多重图问题，并借助Ramsey理论刻画$n$-循环PR盒的激活效应边界。

Fractional chromatic number, maximum degree and girth

We introduce a new method for computing bounds on the independence number and fractional chromatic number of classes of graphs with local constraints, and apply this method in various scenarios. We establish a formula that generates a general upper bound for the fractional chromatic number of triangle-free graphs of maximum degree~$Δ\ge 3$. This upper bound matches that deduced from the fractional version of Reed's bound for small values of~$Δ$, and improves it when~$Δ\ge 17$, transitioning smoothly to the best possible asymptotic regime, barring a breakthrough in Ramsey theory. Focusing on sm…

The early evolution of the H-free process

The H-free process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal H-free graph obtained at the end of the process. When H is strictly 2-balanced, we show that for some c>0, with high probability as $n \to \infty$, the minimum degree in G is at least $cn^{1-(v_H-2)/(e_H-1)}(\log n)^{1/(e_H-1)}$. This gives new lower bounds for the Turán numbers of certain bipartite graphs, such as the compl…

Generalized Paley graphs and their complete subgraphs of orders three and four

Let $k \geq 2$ be an integer. Let $q$ be a prime power such that $q \equiv 1 \pmod {k}$ if $q$ is even, or, $q \equiv 1 \pmod {2k}$ if $q$ is odd. The generalized Paley graph of order $q$, $G_k(q)$, is the graph with vertex set $\mathbb{F}_q$ where $ab$ is an edge if and only if ${a-b}$ is a $k$-th power residue. We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in $G_k(q)$, $\mathcal{K}_4(G_k(q))$, which holds for all $k$. This generalizes the results of Evans, Pulham and Sheehan on the original ($k$=2) Paley …

The Schur degree of additive sets

Let (G, +) be an abelian group. A subset of G is sumfree if it contains no elements x, y, z such that x +y = z. We extend this concept by introducing the Schur degree of a subset of G, where Schur degree 1 corresponds to sumfree. The classical inequality S(n) $\le$ R n (3) -- 2, between the Schur number S(n) and the Ramsey number R n (3) = R(3,. .. , 3), is shown to remain valid in a wider context, involving the Schur degree of certain subsets of G. Recursive upper bounds are known for R n (3) but not for S(n) so far. We formulate a conjecture which, if true, would fill this gap. Indeed, our s…