Ashwin Sah
We improve the upper bound for diagonal Ramsey numbers to \[R(k+1,k+1)\le\exp(-c(\log k)^2)\binom{2k}{k}\] for $k\ge 3$. To do so, we build on a quasirandomness and induction framework for Ramsey numbers introduced by Thomason and extended by Conlon, demonstrating optimal "effective quasirandomness" results about convergence of graphs. This optimality represents a natural barrier to improvement.
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The book graph $B_n^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$. The Ramsey numbers of $B_n^{(k)}$ are known to have strong connections to the classical Ramsey numbers of cliques. Recently, the first author determined the asymptotic order of these Ramsey numbers for fixed $k$, thus answering an old question of Erdős, Faudree, Rousseau, and Schelp. In this paper, we first provide a simpler proof of this theorem. Next, answering a question of the first author, we present a different proof that avoids the use of Szemerédi's regularity lemma, thus providing much tighter …
For a positive integer $r$, let $G(r)$ be the smallest $N$ such that, whenever the edges of the Cartesian product $K_N \times K_N$ are $r$-coloured, then there is a rectangle in which both pairs of opposite edges receive the same colour. In this paper, we improve the upper bounds on $G(r)$ by proving $G(r) \leq \Big(1 - \frac{1}{128}r^{-2}\Big) r^{\binom{r+1}{2}}$, for $r$ large enough. Unlike the previous improvements, which were based on bounds for the size of set systems with restricted intersection sizes, our proof is a form of a quasirandomness argument.
We prove a new upper bound for diagonal two-colour Ramsey numbers, showing that there exists a constant $C$ such that \[r(k+1, k+1) \leq k^{- C \frac{\log k}{\log \log k}} \binom{2k}{k}.\]
We give a coding based perspective, on a result of Erdós, on a lower bound for the diagonal ramsey numbers.
When are off-diagonal hypergraph Ramsey numbers polynomial?
该研究针对拉姆齐理论中3-图非对角超图拉姆齐数何时随n多项式增长的开放问题，证明了紧连通或至多两个紧分支的3-图满足该性质当且仅当它包含于边的迭代膨胀中，推进了相关问题研究。
The book graph $B_n^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$. In the prequel to this paper, we studied the diagonal Ramsey number $r(B_n^{(k)}, B_n^{(k)})$. Here we consider the natural off-diagonal variant $r(B_{cn}^{(k)}, B_n^{(k)})$ for fixed $c \in (0,1]$. In this more general setting, we show that an interesting dichotomy emerges: for very small $c$, a simple $k$-partite construction dictates the Ramsey function and all nearly-extremal colorings are close to being $k$-partite, while, for $c$ bounded away from $0$, random colorings of an appropriate density ar…
Some Combinatorial Formulas Related to Diagonal Ramsey Numbers
本文推导了与对角拉姆齐数 R(k) 相关的一类组合公式，其形式为 F(n,k)=0 当且仅当 n≥R(k)，并与 De Loera 和 Wesley 的近期工作密切相关。
A $k$-uniform loose cycle $\mathcal{C}_n^k$ is a hypergraph with vertex set $\{v_1,v_2,\ldots,v_{n(k-1)}\}$ and with the set of $n$ edges $e_i=\{v_{(i-1)(k-1)+1},v_{(i-1)(k-1)+2},\ldots,v_{(i-1)(k-1)+k}\}$, $1\leq i\leq n$, where we use mod $n(k-1)$ arithmetic. The Ramsey number $R(\mathcal{C}^k_n,\mathcal{C}^k_n)$ is asymptotically $\frac{1}{2}(2k-1)n$ as has been proved by Gyárfás, Sárközy and Szemerédi. In this paper, we investigate to determining the exact value of diagonal Ramsey number of $\mathcal{C}^k_n$ and we show that for $n\geq 2$ and $k\geq 8$ $$R(\mathcal{C}^k_n,\mathcal{C}^k_n)=…
Let the star on $n$ vertices, namely $K_{1,n-1}$ be denoted by $S_n$. If every two coloring of the edges of a complete balanced multipartite graph $K_{j \times s}$ there is a copy of $S_n$ in the first color or a copy of $S_m$ in the second color, then we will say $K_{j \times s}\rightarrow (S_n,S_m)$. The size Ramsey multipartite number $m_j(S_n, S_m)$ is the smallest natural number $s$ such that $K_{j \times s}\rightarrow (S_n,S_m)$. In this paper, we obtain the exact values of the size Ramsey numbers $m_j(S_n,S_m)$ for $n,m \ge 3$ and $j \ge 3$.
Let $R(k_1, \cdots, k_r)$ denote the classical $r$-color Ramsey number for integers $k_i \ge 2$. The Diagonal Conjecture (DC) for classical Ramsey numbers poses that if $k_1, \cdots, k_r$ are integers no smaller than 3 and $k_{r-1} \leq k_r$, then $R(k_1, \cdots, k_{r-2}, k_{r-1}-1, k_r +1) \leq R(k_1, \cdots, k_r)$. We obtain some implications of this conjecture, present evidence for its validity, and discuss related problems. Let $R_r(k)$ stand for the $r$-color Ramsey number $R(k, \cdots, k)$. It is known that $\lim_{r \rightarrow \infty} R_r(3)^{1/r}$ exists, either finite or infinite, the…
An exponential improvement for diagonal Ramsey
本文证明了对角拉姆齐数 $R(k)$ 存在指数改进的上界 $R(k) \leqslant (4 - \varepsilon)^k$，首次突破 Erdős–Szekeres 上界。
For ordered graphs $G$ and $H$, the ordered Ramsey number $r_<(G,H)$ is the smallest $n$ such that every red/blue edge coloring of the complete graph on vertices $\{1,\dots,n\}$ contains either a blue copy of $G$ or a red copy of $H$, where the embedding must preserve the relative order of vertices. One number of interest, first studied by Conlon, Fox, Lee, and Sudakov, is the "off-diagonal" ordered Ramsey number $r_<(M, K_3)$, where $M$ is an ordered matching on $n$ vertices. In particular, Conlon et al. asked what asymptotic bounds (in $n$) can be obtained for $\max r_<(M, K_3)$, where the m…
Some recent results in Ramsey theory
本文综述了Ramsey理论中关于Ramsey数的最新突破，包括对角线、近对角线和多色Ramsey数的指数改进，以及诱导Ramsey数的上界结果。
New bounds for off-diagonal hypergraph Ramsey numbers
该研究针对Conlon等人的拉姆齐数猜想，证明了至多5个紧分支的3-图满足该猜想，推导了其与线性3-图特例的等价性，给出可向前双化3-图的拟多项式上界、非向前层状3-图的指数下界，并分类了链族的拉姆齐增长性。
Diagonal Ramsey numbers for wheels
本文研究了轮图的对角Ramsey数，改进了之前的结果，给出了更精确的上下界，并提供了关于k着色Ramsey数的递归界。
We give an exponential improvement to the lower bound on diagonal Ramsey numbers for any fixed number of colors greater than two.
Turán Colourings in Off-Diagonal Ramsey Multiplicity
本研究将Fox和Wigderson关于由带临界边的连通非3-可着色图加悬挂边构成的图类的Ramsey重数常数由Turán着色达到的结果，推广到涉及两图红、蓝副本加权和最小化的非对角Ramsey重数常数变体。
Bounds on the Threshold Ramsey Multiplicity of Ramsey Numbers with Many Colors
本文推导多色非对角阈值拉姆齐重数的上界，改进了对角二色情形下的基础随机染色估计，得到$m(3,3,3)=5$、$m(3,3,4)\leq10$的结果，并给出通用改进框架。
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。
Retained-Set Descent for Diagonal Ramsey Numbers
本文通过保留一个顶点集并在另一顶点集的蓝色邻域中下降来改进拉姆齐上界，得到对角指数下确界在[1.305,1.307]内，并给出R(k,k)≤3.69507^k的有限推导及最大不动点刻画。
Sharper Zarankiewicz and Diagonal Bipartite Ramsey Bounds
证明二分图边稠密时存在完全二分子图的更优界，改进 Zarankiewicz 与对角二分 Ramsey 数上界至 $O(2^t)$，方法由 GPT-6 Astra 发现。
Off-Diagonal Ramsey Numbers for Linear Hypergraphs
本文研究线性超图的非对角 Ramsey 数，证明在均匀度 $k\ge 4$ 时，存在线性 $k$-均匀超图使 Ramsey 数具有远高于多项式的增长率。
A recent breakthrough of Conlon and Ferber yielded an exponential improvement on the lower bounds for multicolor diagonal Ramsey numbers. In this note, we modify their construction and obtain improved bounds for more than three colors.
The asymptotic of off-diagonal online Ramsey numbers for paths
本文证明路径的在线拉姆齐数下界与最近上界在线性项内匹配，从而确定其渐近值为5/3n，并否定了Cyman等人的猜想。
An Improved Upper Bound for Multicolour Ramsey Numbers
本研究针对多色拉姆齐数，改进了Yang和Mao的相关上界，通过非对角界得到了更优的指数形式上界，推进了拉姆齐数的理论研究。
Off-diagonal Ramsey numbers for slowly growing hypergraphs
本文研究缓慢增长超图的对角外拉姆齐数，证明对任意非k-部缓慢增长k-一致超图F，r(F,n)至少为n^k除以对数因子的量级，并首次确定三元系F_5的对角外拉姆齐数多项式幂次，构造综合运用伪随机图、鞅与超图容器方法。
Off-diagonal Ramsey numbers
通过构造新的无 K_s 图族，改进了非对角 Ramsey 数的下界，几乎匹配已知上界，并证明了当 s, k/s → ∞ 时 r(s,k) = (k/s)^{(1+o(1))s}。
Upper bounds on diagonal Ramsey numbers [after Campos, Griffiths, Morris, and Sahasrabudhe]
本文综述了对角 Ramsey 数上界问题的历史，并介绍了 Campos 等人将上界改进至 $N=3.993^k$ 的突破性证明及后续的替代性概念证明。
Formalising New Mathematics in Isabelle: Diagonal Ramsey
针对证明助手验证全新数学成果案例稀少的问题，本文在Isabelle中完成了2023年公布的对角拉姆齐数新成果的形式化，意外发现该工作高度依赖计算机代数技术，为数学形式化的实际应用提供了新案例。
On off-diagonal $F$-Ramsey numbers
本文研究了非对角$F$-Ramsey数猜想，利用$r(4,t)$的渐近构造证明了当$t_1$为3或4时，$r_{K_s}(t_1,t_2)$与$\inom{r(t_1,t_2)}{s}$在指数项上仅相差低阶项。
On off-diagonal hypergraph Ramsey numbers
研究固定3-一致超图H与完全3-一致超图K_n^{(3)}的拉姆齐数增长速率，证明对奇环链接及多数紧环有下界2^{Ω_H(n log n)}，并构造线性超图H使其拉姆齐数为超多项式，首次分离r(H,K_n^{(3)})与r(H,K_{n,n,n}^{(3)})。
The Ramsey number $r_k(s,n)$ is the minimum $N$ such that every red-blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ contains a red set of size $s$ or a blue set of size $n$, where a set is red (blue) if all of its $k$-subsets are red (blue). A $k$-uniform \emph{tight path} of size $s$, denoted by $P_{s}$, is a set of $s$ vertices $v_1 < \cdots < v_{s}$ in $\mathbb{Z}$, and all $s-k+1$ edges of the form $\{v_j,v_{j+1},\ldots, v_{j + k -1}\}$. Let $r_k(P_s, n)$ be the minimum $N$ such that every red-blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ results in a red $P_{s}$ or a blue …
The $(m,n)$-online Ramsey game is a combinatorial game between two players, Builder and Painter. Starting from an infinite set of isolated vertices, Builder draws an edge on each turn and Painter immediately paints it red or blue. Builder's goal is to force Painter to create either a red $K_m$ or a blue $K_n$ using as few turns as possible. The online Ramsey number $\tilde{r}(m,n)$ is the minimum number of edges Builder needs to guarantee a win in the $(m,n)$-online Ramsey game. By analyzing the special case where Painter plays randomly, we obtain an exponential improvement \[ \tilde{r}(n,n) \…
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(…
Sharp bounds for off-diagonal and tripartite canonical Ramsey numbers
本文研究典型拉姆齐定理的两个变体，给出多部分超图典型拉姆齐数的锐界，并建立非对角典型拉姆齐数在不同参数区间的上下界，包括避免字典序三角形时的上界。
Off-Diagonal Ramsey Multiplicity
本文研究了非对角Ramsey乘数问题，通过引入加权密度求和的概念，解决了几种图对的Ramsey乘数问题。
Off-diagonal online size Ramsey numbers for paths
本文研究路径的非对角在线大小 Ramsey 数，证明 $\tilde{r}(P_k,P_n)\le (5/3+o(1))n$（当 $k=o(n)$），并改进 $\tilde{r}(P_4,P_n)$ 的上界至 $\lceil 7n/5\rceil -1$，验证了相关猜想。
Sharp Diagonal Thresholds for Tight Hamilton Cycles in Uniformly Dense $3$-Graphs
本文在最小顶点度和最小共度数条件下，确定均匀稠密3 - 图中紧密哈密顿圈的尖锐对角阈值，证明\(d > 1/3\)且\(\alpha > f(d)\)时相关密度和度条件迫使存在紧密哈密顿圈，还给出最小共度数条件下阈值，反驳了一些猜想。
We determine the $2$-color Ramsey number of a {\em connected} triangle matching $c(nK_3)$ which is any connected graph containing $n$ vertex disjoint triangles. We obtain that $R(c(nK_3),c(nK_3))=7n-2$, somewhat larger than in the classical result of Burr, Erd\H os and Spencer for a triangle matching, $R(nK_3,nK_3)=5n$. The motivation is to determine the Ramsey number $R(C_n^2,C_n^2)$ of the square of a cycle $C_n^2$. We apply our Ramsey result for connected triangle matchings to show that the Ramsey number of an "almost" square of a cycle $C_n^{2,c}$ (a cycle of length $n$ in which all but at…
A note on the logarithmic exponent of Ramsey number $r(4,t)$
该注记结合非对角拉姆齐数下界与Ajtai等人的独立性猜想证明，得出$r(4,t)$的对数指数相关结果，仅作非正式传阅不投稿。
A New Lower Bound for the Diagonal Poset Ramsey Numbers
本文提出了对角正格Ramsey数R(Q_n,Q_n)的新下界，证明其至少为2.7n加上常数，并探讨了进一步改进的可能性。
Does the $n^{th}$ root of the diagonal Ramsey number converge to a finite limit? The answer is yes. A sequence can be shown to converge if it satifies convergence conditions other than or besides monotonicity. We show such a property holds for which the sequence of $n^{th}$ roots does converge, even if one has no a priori knowledge as to whether the sequence is monotone or not. We show also the $n^{th}$ root of the diagonal Ramsey number can be expressed as a product of two factors, the first being a known convergent sequence and the second being an absolutely convergent infinite series. One a…
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。
Let $K_m^{(3)}$ denote the complete $3$-uniform hypergraph on $m$ vertices and $S_n^{(3)}$ the $3$-uniform hypergraph on $n+1$ vertices consisting of all $\binom{n}{2}$ edges incident to a given vertex. Whereas many hypergraph Ramsey numbers grow either at most polynomially or at least exponentially, we show that the off-diagonal Ramsey number $r(K_{4}^{(3)},S_n^{(3)})$ exhibits an unusual intermediate growth rate, namely, \[ 2^{c \log^2 n} \le r(K_{4}^{(3)},S_n^{(3)}) \le 2^{c' n^{2/3}\log n} \] for some positive constants $c$ and $c'$. The proof of these bounds brings in a novel Ramsey probl…
Polynomial families of incident flags and explicit off-diagonal Ramsey graphs
本文从多项式族构造显式非对角 Ramsey 图，通过插值和标量限制得到指数界，并给出确定性多项式时间算法。
An improved algebraic construction for Ramsey numbers
该研究改进了拉姆齐数的代数构造，给出显式下界，提升了非对角及对角情形的界，得到R(33,t)≥t^{2.1-o(1)}，是首个证明固定s时R(s,t)≥t^c（c>2）的显式构造。
In 1967, Erdős asked for the greatest chromatic number, $f(n)$, amongst all $n$-vertex, triangle-free graphs. An observation of Erdős and Hajnal together with Shearer's classical upper bound for the off-diagonal Ramsey number $R(3, t)$ shows that $f(n)$ is at most $(2 \sqrt{2} + o(1)) \sqrt{n/\log n}$. We improve this bound by a factor $\sqrt{2}$, as well as obtaining an analogous bound on the list chromatic number which is tight up to a constant factor. A bound in terms of the number of edges that is similarly tight follows, and these results confirm a conjecture of Cames van Batenburg, de Jo…
Gyárfás et al. determined the asymptotic value of the diagonal Ramsey number of $\mathcal{C}^k_n$, $R(\mathcal{C}^k_n,\mathcal{C}^k_n),$ generating the same result for $k=3$ due to Haxell et al. Recently, the exact values of the Ramsey numbers of 3-uniform loose paths and cycles are completely determined. These results are motivations to conjecture that for every $n\geq m\geq 3$ and $k\geq 3,$ $$R(\mathcal{C}^k_n,\mathcal{C}^k_m)=(k-1)n+\lfloor\frac{m-1}{2}\rfloor,$$ as mentioned by Omidi et al. More recently, it is shown that this conjecture is true for $n=m\geq 2$ and $k\geq 7$ and for $k=4$…
For two graphs $G^<$ and $H^<$ with linearly ordered vertex sets, the ordered Ramsey number $r_<(G^<,H^<)$ is the minimum $N$ such that every red-blue coloring of the edges of the ordered complete graph on $N$ vertices contains a red copy of $G^<$ or a blue copy of $H^<$. For a positive integer $n$, a nested matching $NM^<_n$ is the ordered graph on $2n$ vertices with edges $\{i,2n-i+1\}$ for every $i=1,\dots,n$. We improve bounds on the ordered Ramsey numbers $r_<(NM^<_n,K^<_3)$ obtained by Rohatgi, we disprove his conjecture by showing $4n+1 \leq r_<(NM^<_n,K^<_3) \leq (3+\sqrt{5})n$ for eve…
Sharper Ramsey lower bounds from refined Gaussian estimates
本文通过精细的累积量生成函数界改进了对角拉姆齐数$R(\ell,C\ell)$的指数下界，对每个固定的$C>1$实现了严格正的指数增益。
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…
The Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there are $s$ integers such that every $k$-tuple among them is red, or $n$ integers such that every $k$-tuple among them is blue. We prove the following new lower bounds for 4-uniform hypergraph Ramsey numbers: $$r_4(5,n) > 2^{n^{c\log n}} \qquad \hbox{ and } \qquad r_4(6,n) > 2^{2^{cn^{1/5}}},$$ where $c$ is an absolute positive constant. This substantially improves the previous best bounds of $2^{n^{c\log\log n}}$ and $2^{n^{c\log n}}$, respectively. Using previously …
Ramsey-Turán Problems with small independence numbers
本文研究了$H=K_s$（$s\leq 13$）时的Ramsey-Turán数，通过应用Szemerédi正则引理和相关随机选择方法，证明了当独立数约束$f(n)$接近非对角Ramsey数逆函数时团存在相变现象。
On the off-diagonal unordered Erdős-Rado numbers
本文研究无序非对角规范 Ramsey 数，证明 CR(s,r) 的新界并匹配已知下界，同时给出 ER(m,ℓ,r) 的界。
Diagonal poset Ramsey numbers
本文研究布尔格上的对角偏序集拉姆齐数，将 $R(Q_n,Q_n)$ 的上界改进为 $n^2-(1-o(1))n\log n$，并确定 $n$-叉和 $n$-菱形的对角拉姆齐数（误差至多2），同时证明弱版本上界 $0.96n^2$。
Ramsey numbers of grid graphs
本文研究了网格图的非对角Ramsey数，证明了除$G_{2\times 2}$外的任意环及递归膨胀的网格子图，其Ramsey数均为多项式级别，并基于多色Erdős-Hajnal猜想给出了两行网格子图的多项式上界。
The poset Ramsey number $R(Q_m,Q_n)$ is the smallest integer $N$ such that any blue-red coloring of the elements of the Boolean lattice $Q_N$ has a blue induced copy of $Q_m$ or a red induced copy of $Q_n$. The weak poset Ramsey number $R_w(Q_m,Q_n)$ is defined analogously, with weak copies instead of induced copies. It is easy to see that $R(Q_m,Q_n) \ge R_w(Q_m,Q_n)$. Axenovich and Walzer showed that $n+2 \le R(Q_2,Q_n) \le 2n+2$. Recently, Lu and Thompson improved the upper bound to $\frac{5}{3}n+2$. In this paper, we solve this problem asymptotically by showing that $R(Q_2,Q_n)=n+O(n/\log …
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…
Book Ramsey numbers via algebraic constructions
本文通过构造新的强正则图族，证明了无穷多个n满足R(B_n,B_n)=4n+1，并改进了R(B_{n-2},B_n)的上界，去除了原有模条件。
A note on multicolour Ramsey numbers and random sphere graphs
本文通过将二项随机图替换为随机球图，改进了多色Ramsey数的下界，实现了指数级的提升。
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