Two multicolor Ramsey numbers

An improved lower bound for multicolor Ramsey numbers and the half-multiplicity Ramsey number problem

The multicolor Ramsey number problem asks, for each pair of natural numbers $\ell$ and $t$, for the largest $\ell$-coloring of a complete graph with no monochromatic clique of size $t$. Recent works of Conlon-Ferber and Wigderson have improved the longstanding lower bound for this problem. We make a further improvement by replacing an explicit graph appearing in their constructions by a random graph. Graphs useful for this construction are exactly those relevant for a problem of Erdős on graphs with no large cliques and few large independent sets. We also make some basic observations about thi…

On multicolor Ramsey number for 3-paths of length three

We show that if we color the hyperedges of the complete $3$-uniform complete graph on $2n+\sqrt{18n+1}+2$ vertices with $n$ colors, then one of the color classes contains a loose path of length three.

多色Ramsey下界更新

An update on multicolor Ramsey lower bounds

基于球面随机几何图构造，改进了多色Ramsey数的下界，提供了更精确的理论结果。

On multicolor Ramsey numbers for loose $k$-paths of length three

We show that there exists an absolute constant $A$ such that for each $k\ge2$ and every coloring of the edges of the complete $k$-uniform hypergraph on $ Ar$ vertices with $r$ colors, one of the color classes contains a loose path of length three.

New upper bound for multicolor Ramsey number of odd cycles

Let $r_k(C_{2m+1})$ be the $k$-color Ramsey number of an odd cycle $C_{2m+1}$ of length $2m+1$. It is shown that for each fixed $m\ge2$, \[r_k(C_{2m+1})<c^{k}\sqrt{k!}\] for all sufficiently large $k$, where $c=c(m)>0$ is a constant. This improves an old result by Bondy and Erdős (Ramsey numbers for cycles in graphs, J. Combin. Theory Ser. B 14 (1973) 46-54).

Multicolor list Ramsey numbers grow exponentially

The list Ramsey number $R_{\ell}(H,k)$, recently introduced by Alon, Bucić, Kalvari, Kuperwasser, and Szabó, is a list-coloring variant of the classical Ramsey number. They showed that if $H$ is a fixed $r$-uniform hypergraph that is not $r$-partite and the number of colors $k$ goes to infinity, $e^{Ω(\sqrt{k})} \le R_{\ell} (H,k) \le e^{O(k)}$. We prove that $R_{\ell}(H,k) = e^{Θ(k)}$ if and only if $H$ is not $r$-partite.

On multicolor Ramsey numbers of triple system paths of length 3

Let $\mathcal{H}$ be a 3-uniform hypergraph. The multicolor Ramsey number $ r_k(\mathcal{H})$ is the smallest integer $n$ such that every coloring of $ \binom{[n]}{3}$ with $k$ colors has a monochromatic copy of $\mathcal{H}$. Let $ \mathcal{L}$ be the loose 3-uniform path with 3 edges and $ \mathcal{M}$ denote the messy 3-uniform path with 3 edges; that is, let $\mathcal{L} = \{abc, cde, efg\}$ and $\mathcal{M} = \{ abc, bcd, def\}$. In this note we prove $ r_k(\mathcal{L}) < 1.54k$ and $ r_k(\mathcal{M}) < 1.6k$ for $k$ sufficiently large.

多色星临界拉姆齐数的下界

Lower Bounds for Multicolor Star-Critical Ramsey Numbers

本文研究多色星临界拉姆齐数，给出其消失的等价判据及非消失时的通用下界，并计算了 $r_*(P_k, P_3, P_3)$，同时证明 $r_*(C_5, P_3)=3$。

多色Ramsey数的新上界

New upper bound for multicolor Ramsey numbers

本研究针对多色Ramsey数上界问题，结合可变阶正系数根滤波器与多色书方法的保留脊细化技术，证明了满足一定参数条件的对角r色图Ramsey数新上界公式。

Improved bounds on the multicolor Ramsey numbers of paths and even cycles

We study the multicolor Ramsey numbers for paths and even cycles, $R_k(P_n)$ and $R_k(C_n)$, which are the smallest integers $N$ such that every coloring of the complete graph $K_N$ has a monochromatic copy of $P_n$ or $C_n$ respectively. For a long time, $R_k(P_n)$ has only been known to lie between $(k-1+o(1))n$ and $(k + o(1))n$. A recent breakthrough by Sárközy and later improvement by Davies, Jenssen and Roberts give an upper bound of $(k - \frac{1}{4} + o(1))n$. We improve the upper bound to $(k - \frac{1}{2}+ o(1))n$. Our approach uses structural insights in connected graphs without a l…

On the multicolor Ramsey number of a graph with m edges

The multicolor Ramsey number $r_k(F)$ of a graph $F$ is the least integer $n$ such that in every coloring of the edges of $K_n$ by $k$ colors there is a monochromatic copy of $F$. In this short note we prove an upper bound on $r_k(F)$ for a graph $F$ with $m$ edges and no isolated vertices of the form $k^{6km^{2/3}}$ addressing a question of Sudakov [ Adv. Math. 227 (2011), no. 1, 601--609]. Furthermore, the constant in the exponent in the case of bipartite $F$ and two colors is lowered so that $r_2(F)\le 2^{(1+o(1))2\sqrt{2m}}$ improving the result of Alon, Krivelevich and Sudakov [Combin. Pr…

Multicolor Ramsey Numbers for Complete Bipartite Versus Complete Graphs

Let H_1, ..., H_k be graphs. The multicolor Ramsey number r(H_1,...,H_k) is the minimum integer r such that in every edge-coloring of K_r by k colors, there is a monochromatic copy of H_i in color i for some 1 <= i <= k. In this paper, we investigate the multicolor Ramsey number $r(K_{2,t},...,K_{2,t},K_m)$, determining the asymptotic behavior up to a polylogarithmic factor for almost all ranges of t and m. Several different constructions are used for the lower bounds, including the random graph and explicit graphs built from finite fields. A technique of Alon and Rödl using the probabilistic …

奇环的多色拉姆齐数是超指数级的

Multicolor Ramsey numbers of odd cycles are superexponential

该研究对 OpenAI 关于三角形多色拉姆齐数的递归构造作出修改，证明固定奇环的多色拉姆齐数呈超指数增长，相关证明由 ChatGPT 5.6 Pro/Sol 自主完成。

Multicolor Ramsey numbers via pseudorandom graphs

A weakly optimal $K_s$-free $(n,d,λ)$-graph is a $d$-regular $K_s$-free graph on $n$ vertices with $d=Θ(n^{1-α})$ and spectral expansion $λ=Θ(n^{1-(s-1)α})$, for some fixed $α>0$. Such a graph is called optimal if additionally $α= \frac{1}{2s-3}$. We prove that if $s_{1},\ldots,s_{k}\ge3$ are fixed positive integers and weakly optimal $K_{s_{i}}$-free pseudorandom graphs exist for each $1\le i\le k$, then the multicolor Ramsey numbers satisfy \[ Ω\Big(\frac{t^{S+1}}{\log^{2S}t}\Big)\le r(s_{1},\ldots,s_{k},t)\le O\Big(\frac{t^{S+1}}{\log^{S}t}\Big), \] as $t\rightarrow\infty$, where $S=\sum_{i…

若干小型多色Ramsey数的新界

New bounds for some small multicolor Ramsey numbers

该研究利用SAT求解器搜索结构化着色，推导了两组多色Ramsey数的下界，收紧了环相关多色Ramsey数的上界并确定两组精确值，还枚举了两类含星图的Ramsey数的临界图。

多色有序拉姆齐数的一个一般上界

A General Upper Bound on Multicolor Ordered Ramsey Numbers

本文给出了多色有序拉姆齐数的一般上界，基于区间色数和退化度，扩展了先前结果，并证明对有序匹配几乎紧。

二元域上的多色向量空间拉姆齐数

Multicolor vector space Ramsey numbers over the binary field

研究二元域上多色向量空间拉姆齐数\(R_2(t;k)\)，通过二元单纯码将问题简化为经典超图拉姆齐问题，给出\(R_2(t;k)\)上界，改进了此前结果，还得到二元射影空间色数下界，\(t = 2\)时恢复与三角形多色拉姆齐数联系。

四边形的多色、多部拉姆齐数

Multicolor, multipartite Ramsey numbers for quadrilateral

本文研究四边形的多色多部拉姆齐数，给出一般上界及二色情形的全部精确值，并确定三部四色拉姆齐数为11。

极小多色 Ramsey 图最小度的改进界

Improved bounds for the minimum degree of minimal multicolor Ramsey graphs

本文通过结合代数与概率着色方案，构造了 $r$ 个边不相交的 $K_{k+1}$-free 图，改进了极小多色 Ramsey 图最小度的上界。

3-均匀单调路径与多色拉姆齐数

3-uniform monotone paths and multicolor Ramsey numbers

本文研究3-均匀单调路径超图，建立三角形多色拉姆齐数与相应超图拉姆齐数之间的指数级上下界关系，从而将Erdős关于三角形多色拉姆齐数是否指数增长的经典问题等价转化为超图拉姆齐数的指数增长问题。

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…

Multicolor Gallai-Ramsey numbers of $C_9$ and $C_{11}$

A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai $k$-coloring is a Gallai coloring that uses $k$ colors. We study Ramsey-type problems in Gallai colorings. Given an integer $k\ge1$ and a graph $H$, the Gallai-Ramsey number $GR_k(H)$ is the least positive integer $n$ such that every Gallai $k$-coloring of the complete graph on $n$ vertices contains a monochromatic copy of $H$. It turns out that $GR_k(H)$ is more well-behaved than the classical Ramsey number $R_k(H)$. However, finding exact values of $GR_k (H)$ is far from trivial. In this …

多色拉姆齐数 $r_k(K_{2, t + 1})$ 的一个新下界

A new lower bound for the multicolor Ramsey number $r_k(K_{2, t + 1})$

本文通过构造素数幂 $t$ 下的无 $K_{2,t+1}$ 图并用于边划分，改进了 $k$ 与 $t$ 为同一素数幂时多色拉姆齐数 $r_k(K_{2,t+1})$ 的下界，得到 $tk^2+1$ 的下界。

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…

多重色Ramsey数在3-一致超图中的增长速率

The growth rate of multicolor Ramsey numbers of $3$-graphs

本文研究了3-一致超图的多重色Ramsey数在颜色数趋于无穷时的增长行为，确定了其塔高并给出了多项式、指数或双指数增长的具体条件，回答了Axenovich等人提出的问题。

Multicolor Ramsey numbers for triple systems

Given an $r$-uniform hypergraph $H$, the multicolor Ramsey number $r_k(H)$ is the minimum $n$ such that every $k$-coloring of the edges of the complete $r$-uniform hypergraph $K_n^r$ yields a monochromatic copy of $H$. We investigate $r_k(H)$ when $k$ grows and $H$ is fixed. For nontrivial 3-uniform hypergraphs $H$, the function $r_k(H)$ ranges from $\sqrt{6k}(1+o(1))$ to double exponential in $k$. We observe that $r_k(H)$ is polynomial in $k$ when $H$ is $r$-partite and at least single-exponential in $k$ otherwise. Erdős, Hajnal and Rado gave bounds for large cliques $K_s^r$ with $s\ge s_0(r)…

A note on multicolor Ramsey number of small odd cycles versus a large clique

Let $R_k(H;K_m)$ be the smallest number $N$ such that every coloring of the edges of $K_{N}$ with $k+1$ colors has either a monochromatic $H$ in color $i$ for some $1\leqslant i\leqslant k$, or a monochromatic $K_{m}$ in color $k+1$. In this short note, we study the lower bound for $R_k(H;K_m)$ when $H$ is $C_5$ or $C_7$, respectively. We show that \begin{equation*} R_{k}(C_5;K_m)=Ω(m^{\frac{3k}{8}+1}/(\log{m})^{\frac{3k}{8}+1}), \end{equation*} and \begin{equation*} R_{k}(C_7;K_m)=Ω(m^{\frac{2k}{9}+1}/(\log{m})^{\frac{2k}{9}+1}), \end{equation*} for fixed positive integer $k$ and $m\rightarro…

有序匹配的多色 Ramsey 数

Multicolor Ramsey numbers of ordered matchings

本文否定回答了有序匹配多色Ramsey数的多项式对数上界猜想，证明几乎每个完美匹配的下界指数为$q$次对数，并给出两个应用：证明有界度无环有向图的下界猜想和加强$M$-自由有序图色数下界。

Multicolor Size-Ramsey Number of Cycles

Given a positive integer $ r $, the $ r $-color size-Ramsey number of a graph $ H $, denoted by $ \hat{R}(H, r) $, is the smallest integer $ m $ for which there exists a graph $ G $ with $ m $ edges such that, in any edge coloring of $ G $ with $ r $ colors, $G$ contains a monochromatic copy of $ H $. Haxell, Kohayakawa and Łuczak showed that the size-Ramsey number of a cycle $ C_n $ is linear in $ n $ i.e. $ \hat{R}(C_n, r) \leq c_rn $, for some constant $ c_r $. Their proof, however, is based on the Szemerédi's regularity lemma and so no specific constant $ c_r $ is known. Javadi, Khoeini, O…

关于 Ramsey 理论中非对称团组发送者的使用

On the use of senders for asymmetric tuples of cliques in Ramsey theory

本文针对任意团组构造了信号发送者和决定者工具图，并利用这些工具将该领域的三个经典定理推广至非对称多色 Ramsey 设置。

双星的多色二部Ramsey数

Multicolor bipartite Ramsey number of double stars

本文研究了双星$S(n,m)$的多色二部Ramsey数，通过在二部设定中应用Turán论证，证明了在$k=2$且$n\ge m$或$k\ge3$且$n\ge 2m$条件下，该Ramsey数等于$kn+1$。

星形树的multicolor Ramsey和list Ramsey数

Multicolor Ramsey and list Ramsey numbers for star-like trees

本文通过统一直接证明确认了Alon等人关于星形图list Ramsey数的猜想，并确定了双星和细分星在显式参数下的Ramsey数，填补了已知界的空白。

Sub-Ramsey numbers for arithmetic progressions

Let the integers $1,\ldots,n$ be assigned colors. Szemerédi's theorem implies that if there is a dense color class then there is an arithmetic progression of length three in that color. We study the conditions on the color classes forcing totally multicolored arithmetic progressions of length 3. Let $f(n)$ be the smallest integer $k$ such that there is a coloring of $\{1, \ldots, n\}$ without totally multicolored arithmetic progressions of length three and such that each color appears on at most $k$ integers. We provide an exact value for $f(n)$ when $n$ is sufficiently large, and all extremal…

Online and size anti-Ramsey numbers

A graph is properly edge-colored if no two adjacent edges have the same color. The smallest number of edges in a graph any of whose proper edge colorings contains a totally multicolored copy of a graph H is the size anti-Ramsey number AR_s(H) of H. This number in offline and online setting is investigated here.

二分图长细分的多色大小拉姆齐数

The Multicolor Size-Ramsey Number of Bipartite Long Subdivisions

研究了二分图长细分的多色大小拉姆齐数，通过不同方法改进了相关界。

奇围长的Erdős-Hajnal猜想

Locally bipartite subgraphs via multicolor Ramsey numbers

该研究针对Erdős-Hajnal猜想，改进了$f_4(k)$的上界，还证明了奇数$g\ge5$时对应函数$h_g(k)$的增长性，证实了Mohar和Wu2018年的相关猜想。

多重颜色诱导大小拉姆齐数的长细分

The multicolor induced size-Ramsey number of long subdivisions

本文研究了长细分的多重颜色诱导大小拉姆齐数，证明了在特定条件下，该数与图的结构和参数有关，并给出了改进的界。

星形图与路径图的多色 Ramsey 数

Multicolor Ramsey numbers on stars versus pat

本文研究多颗星形图与一条路径图的多色 Ramsey 数，在特定模同余条件下给出新下界与精确值，部分证实 Wang 的猜想。

双星的多色拉姆齐数

Multicolor Ramsey Number for Double Stars

本文研究了双星图的多色拉姆齐数，证明了在特定条件下其值为kn+m+2，并探讨了列表拉姆齐数的相关问题。

有向图的 Mathon 型构造与 Ramsey 数的改进下界

A Mathon-type construction for digraphs and improved lower bounds for Ramsey numbers

本文提出有向图的 Mathon 型构造，证明其与 k 次幂 Paley 有向图的联系，并据此改进多色有向 Ramsey 数的下界。

On some three color Ramsey numbers for paths, cycles, stripes and stars

For given graphs $G_{1}, G_{2}, ... , G_{k}, k \geq 2$, the multicolor Ramsey number $R(G_{1}, G_{2}, ... , G_{k})$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph of order $n$ with $k$ colors, then it always contains a monochromatic copy of $G_{i}$ colored with $i$, for some $1 \leq i \leq k$. The bipartite Ramsey number $b(G_1, \cdots, G_k)$ is the least positive integer $b$ such that any coloring of the edges of $K_{b,b}$ with $k$ colors will result in a monochromatic copy of bipartite $G_i$ in the $i$-th color, for some $i$, $1 \le i \le k$. Th…

关于连通图多色大小Ramsey数的注记

A note on the multicolor size-Ramsey numbers of connected graphs

本文研究连通图的多色大小Ramsey数，证明非星形连通图的该数为Ω(r²m)，并加强了树的结论，指出非星形树的Ramsey数与β(T)相关。

On Some Multicolor Ramsey Numbers Involving $K_3+e$ and $K_4-e$

The Ramsey number $R(G_1, G_2, G_3)$ is the smallest positive integer $n$ such that for all 3-colorings of the edges of $K_n$ there is a monochromatic $G_1$ in the first color, $G_2$ in the second color, or $G_3$ in the third color. We study the bounds on various 3-color Ramsey numbers $R(G_1, G_2, G_3)$, where $G_i \in \{K_3, K_3+e, K_4-e, K_4\}$. The minimal and maximal combinations of $G_i$'s correspond to the classical Ramsey numbers $R_3(K_3)$ and $R_3(K_4)$, respectively, where $R_3(G) = R(G, G, G)$. Here, we focus on the much less studied combinations between these two cases. Through co…

Linear Ramsey numbers for bounded-degree hypergraphs

We show that the Ramsey number is linear for every uniform hypergraph with bounded-degree. This is a hypergraph extension of the famous theorem for ordinary graphs which Chvátal et al. showed in 1983. Our proof is simple, contains the multicolor case, and provides a strong embedding lemma. It shows the potential of a new hypergraph regularity lemma by the author.

On three-color Ramsey number of paths

Let $G_1, G_2, ..., G_t$ be graphs. The multicolor Ramsey number $R(G_1, G_2, ..., G_t)$ is the smallest positive integer $n$ such that if the edges of complete graph $K_n$ are partitioned into $t$ disjoint color classes giving $t$ graphs $H_1,H_2,...,H_t$, then at least one $H_i$ has a subgraph isomorphic to $G_i$. In this paper, we prove that if $(n,m)\neq (3,3), (3,4)$ and $m\geq n$, then $R(P_3,P_n,P_m)=R(P_n,P_m)=m+\lfloor \frac{n}{2}\rfloor-1$. Consequently $R(P_3,mK_2,nK_2)=2m+n-1$ for $m\geq n\geq 3$.

Boolean lattices: Ramsey properties and embeddings

A subposet $Q'$ of a poset $Q$ is a copy of a poset $P$ if there is a bijection $f$ between elements of $P$ and $Q'$ such that $x\leq y$ in $P$ iff $f(x)\leq f(y)$ in $Q'$. For posets $P, P'$, let the poset Ramsey number $R(P,P')$ be the smallest $N$ such that no matter how the elements of the Boolean lattice $Q_N$ are colored red and blue, there is a copy of $P$ with all red elements or a copy of $P'$ with all blue elements. We provide some general bounds on $R(P,P')$ and focus on the situation when $P$ and $P'$ are both Boolean lattices. In addition, we give asymptotically tight bounds for t…

优化Ramsey数的CGMS上界

Optimizing the CGMS upper bound on Ramsey numbers

本文简化了Campos等人关于Ramsey数上界的突破性证明，用归纳陈述替代书算法，优化参数后得到对角Ramsey数上界$R(k,k) \leq (3.8)^{k+o(k)}$，还将非对角Ramsey数上界推广到多色情形。

通过鲁棒或多项式实现施里弗数拟张量积与多色拉姆齐界

Schrijver Number Quasi-Tensorization and Multicolor Ramsey Bounds via Robust OR Polynomials

研究通过鲁棒或多项式框架，在无锐角族和多色拉姆齐数两个应用中，证明施里弗数拟张量积结果及改进几何引理\(r\)依赖性，得出无锐角集大小界和新的多色拉姆齐数上界。

The Ramsey number of a long cycle and complete graphs

In this paper, we prove that the multicolored Ramsey number $R(G_1,\dots,G_n,K_{n_1},\dots,K_{n_r})$ is at least $(γ-1)(κ-1)+1$ for arbitrary connected graphs $G_1,\dots,G_n$ and $n_1,\dots,n_r\in\mathbb{N}$, where $γ=R(G_1,\dots,G_n)$ and $κ=R(K_{n_1},\dots,K_{n_r})$. Erd\H os at al. conjectured that $R(C_n,K_l)=(n-1)(l-1)+1$ for every $n\geq l\geq 3$ except for $n=l=3$. Nikiforov proved this conjecture for $n\geq 4l+2$. Using the above bound, we derive the following generalization of this result. $R(C_n,K_{n_1},\dots,K_{n_r})=(n-1)(κ-1)+1$, where $κ=R(K_{n_1},\dots,K_{n_r})$ and $n\geq 4κ+2$…

Set-coloring Ramsey numbers via codes

For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number $R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph $K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is guaranteed to be a monochromatic clique on $n$ vertices, that is, a subset of $n$ vertices where all of the edges between them receive a common color. In particular, the case $s=1$ corresponds to the classical multicolor Ramsey number. We prove general upper and lower bounds on $R(n;r,s)$ which imply that $R(n;r,s) = 2^{Θ(nr)}$ if $s/r$ is bounded away from $0$ and $1…

二部 Ramsey 数 $BR(C_8, C_{2n})$

The bipartite Ramsey numbers $BR(C_8, C_{2n})$

本文研究二部 Ramsey 数，针对任意二部图定义多色二部 Ramsey 数，并给出了当 $n\geq2$ 时 $BR(C_8,C_{2n})$ 的精确值。

超图中路径的多色size-Ramsey数的下界

A lower bound on the multicolor size-Ramsey numbers of paths in hypergraphs

该研究针对固定k、r递增的k一致紧路径多色size-Ramsey数，得出Ω_k(r^k n)的下界，推广了图论中的已知结果，并将结论拓展到ℓ-重叠路径，改进了k=3时的2色下界。

Zarankiewicz Numbers and Bipartite Ramsey Numbers

The Zarankiewicz number $z(b;s)$ is the maximum size of a subgraph of $K_{b,b}$ which does not contain $K_{s,s}$ as a subgraph. The two-color bipartite Ramsey number $b(s,t)$ is the smallest integer $b$ such that any coloring of the edges of $K_{b,b}$ with two colors contains a $K_{s,s}$ in the first color or a $K_{t,t}$ in the second color. In this work, we design and exploit a computational method for bounding and computing Zarankiewicz numbers. Using it, we obtain several new values and bounds on $z(b;s)$ for $3 \le s \le 6$. Our approach and new knowledge about $z(b;s)$ permit us to improv…

关于平衡双星的多色拉姆齐数

On the multicolor Ramsey numbers of balanced double stars

本文研究平衡双星$S_{n,n}$的多色拉姆齐数与二部多色拉姆齐数，推导得到$r\neq3$时两类拉姆齐数的上下界，并改进了$r=3$、$r=5$时二部拉姆齐数的下界。