A New Upper Bound for Diagonal Ramsey Numbers
David Conlon
本文证明了对角双色Ramsey数的一个新的上界，证明了存在一个常数$C$，使得\[r(k+1, k+1) \leq k^{- C \frac{\log k}{\log \log k}} \binom{2k}{k}.\]
We prove a new upper bound for diagonal two-colour Ramsey numbers, showing that there exists a constant $C$ such that \[r(k+1, k+1) \leq k^{- C \frac{\log k}{\log \log k}} \binom{2k}{k}.\]
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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.
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。
Diagonal Ramsey numbers of loose cycles in uniform hypergraphs
一个$k$ -一致松圈$\mathcal{C}_n^k$是一个顶点集为$\{v_1,v_2,\ldots,v_{n(k-1)}\}$，边集为$e_i=\{v_{(i-1)(k-1)+1},v_{(i-1)(k-1)+2},\ldots,v_{(i-1)(k-1)+k}\}$，$1\leq i\leq n$的超图，这里我们使用模$n(k-1)$算法. Ramsey数$R(\mathcal{C}^k_n,\mathcal{C}^k_n)$是渐近的$\frac{1}{2}(2k-1)n$，正如Gyárfás、Sárközy和Szemerédi所证明的。本文研究了$\mathcal{C}^k_n$的对角Ramsey数的确定问题，证明了对于$n\geq 2$和$k\geq 8$ $$R(\mathcal{C}^k_n,\mathcal{C}^k_n)=(k-1)n+\lfloor\frac{n-1}{2}\rfloor.$$
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。
Recursive upper bounds for the vertex online Ramsey game with applications to hypergraph Ramsey numbers
本文研究了超图Ramsey数的递归上界，提出改进的递推关系并应用于超图Ramsey数的上界优化。
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 fan $F_n$ is a graph consisting of $n$ triangles, all having precisely one common vertex. Currently, the best known bounds for the Ramsey number $R(F_n)$ are $9n/2-5 \leq R(F_n) \leq 11n/2+6$, obtained by Chen, Yu and Zhao. We improve the upper bound to $31n/6+O(1)$.
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。
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).
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。
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。
We consider following geometric Ramsey problem: find the least dimension $n$ such that for any 2-coloring of edges of complete graph on the points $\{\pm 1\}^n$ there exists 4-vertex coplanar monochromatic clique. Problem was first analyzed by Graham and Rothschild and they gave an upper bound: $n\le F(F(F(F(F(F(F(12)))))))$, where $F(m) = 2\uparrow^m3$. In 2014 Lavrov, Lee and Mackey greatly improved this result by giving upper bound $n< 2\uparrow\uparrow\uparrow 6 < F(5)$. In this paper we revisit their estimates and reduce upper bound to $n< 2\uparrow\uparrow\uparrow 5$
Since 2002, the best known upper bound on the Ramsey numbers R n (3) = R(3,. .. , 3) is R n (3) $\le$ n!(e -- 1/6) + 1 for all n $\ge$ 4. It is based on the current estimate R 4 (3) $\le$ 62. We show here how any closing-in on R 4 (3) yields an improved upper bound on R n (3) for all n $\ge$ 4. For instance, with our present adaptive bound, the conjectured value R 4 (3) = 51 implies R n (3) $\le$ n!(e -- 5/8) + 1 for all n $\ge$ 4.
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 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}。
The two-colour Ramsey number $R(m,n)$ is the least natural number $p$ such that any graph of order $p$ must contain either a clique of size $m$ or an independent set of size $n$. We exhibit a method for computing upper bounds for $R(m,n)$ recursively, using known upper bounds of $R(\cdot,\cdot)$ with lower values for at least one of the arguments. We also give an example of how this method could be used to improve several of the best known bounds that are available in the literature (which however soon will be obsolete due to a forthcoming work).
A $k$-ordering of a graph $G$ assigns distinct order-labels from the set $\{1,\ldots,|G|\}$ to $k$ vertices in $G$. Given a $k$-ordering $H$, the ordered Ramsey number $R_<(H)$ is the minimum $n$ such that every edge-2-coloring of the complete graph on the vertex set $\{1, \ldots, n\}$ contains a copy of $H$, the $i$th smallest vertex of which either has order-label $i$ in $H$ or no order-label in $H$. This paper conducts the first systematic study of ordered Ramsey numbers for $1$-orderings of small graphs. We provide upper bounds for $R_<(H)$ for each connected $1$-ordering $H$ on $4$ vertic…
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。
The restricted $(m,n;N)$-online Ramsey game is a game played between two players, Builder and Painter. The game starts with $N$ isolated vertices. Each turn Builder picks an edge to build and Painter chooses whether that edge is red or blue, and Builder aims to create a red $K_m$ or blue $K_n$ in as few turns as possible. The restricted online Ramsey number $\tilde{r}(m,n;N)$ is the minimum number of turns that Builder needs to guarantee her win in the restricted $(m,n;N)$-online Ramsey game. We show that if $N=r(n,n)$, \[ \tilde{r}(n,n;N)\le \binom{N}{2} - Ω(N\log N), \] motivated by a questi…
The study of upper density problems on Ramsey theory was initiated by Erdős and Galvin in 1993. In this paper we are concerned with the following problem: given a fixed finite graph $F$, what is the largest value of $λ$ such that every 2-edge-coloring of the complete graph on $\mathbb{N}$ contains a monochromatic infinite $F$-factor whose vertex set has upper density at least $λ$? Here we prove a new lower bound for this problem. For some choices of $F$, including cliques and odd cycles, this new bound is sharp, as it matches an older upper bound. For the particular case where $F$ is a triangl…
For a fixed infinite graph $H$, we study the largest density of a monochromatic subgraph isomorphic to $H$ that can be found in every two-coloring of the edges of $K_{\mathbb{N}}$. This is called the Ramsey upper density of $H$, and was introduced by Erdős and Galvin. Recently, the Ramsey upper density of the infinite path was determined. Here, we find the value of this density for all locally finite graphs $H$ up to a factor of 2, answering a question of DeBiasio and McKenney. We also find the exact density for a wide class of bipartite graphs, including all locally finite forests. Our approa…
An improved upper bound for the multicolour Ramsey number of odd cycles
该研究推导出长度为$2\ell+1$的奇环的$k$色Ramsey数的改进上界$(4\ell)^k \cdot k^{k/\ell}$，证明了Fox的猜想，实现了自1973年以来该问题指数项的首次超绝对常数因子改进。
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。
Three Proofs of the Hypergraph Ramsey Theorem (An exposition)
Ramsey，Erdos-Rado和Conlon-Fox-Sudakov给出了3-超图Ramsey定理的证明，并给出了3-超图Ramsey数的越来越好的上界. Ramsey和Erdos-Rado也证明了a-超图Ramsey定理。Conlon-Fox-Sudakov注意到他们在3-超图Ramsey数上的上界，以及Erdos-Rado的递归（这是Erdos-Rado证明的关键），产生了a-超图Ramsey数的改进边界。我们提出了所有这些证明和状态明确的界限为2色的情况下和C色的情况下。我们给出了一个更详细的分析Conlon-Fox-Sudakov的结构，从而得到一个稍微好一点的界限。
Upper bounds for ordered Ramsey numbers of forests and bounded-degree graphs
本文证明了有序森林和固定有界度图的有序Ramsey数的两个上界，分别回答了Geneson等人的问题并给出了多项式界。
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)})。
Off-diagonal hypergraph Ramsey numbers
拉姆齐数$r_k(s,n)$是最小的$N$，使得$\{1, \ldots, N\}$的$k$ -子集的每个红-蓝着色包含大小为$s$的红色集合或大小为$n$的蓝色集合，其中如果集合的所有$k$ -子集都是红色（蓝色），则集合是红色（蓝色）。由$P_{s}$表示的大小为$s$的$k$ -均匀\emph{tight path}是$\mathbb{Z}$中的$s$顶点$v_1 < \cdots < v_{s}$和形式为$\{v_j,v_{j+1},\ldots, v_{j + k -1}\}$的所有$s-k+1$边的集合。令$r_k(P_s, n)$为最小$N$，使得$\{1, \ldots, N\}$的$k$ -子集的每个红-蓝着色导致红色$P_{s}$或大小为$n$的蓝色集合。估计$k=2$的$r_k(s,n)$和$r_k(P_s, n)$的问题可以追溯到1935年Erdos和Szekeres的开创性工作，而$k\ge 3$的情况是由Erdos和Rado在1952年首次研究的。在本文中，我们推导出一个定量关系之间的$r_k(P_s, n)$和$r_k(n, n)$的双变异。这产生几个结果，包括以下：（1）我们确定正确的塔增长率为$r_k(s,n)$和$r_k(P_s, n)$为$s \ge k+3$。Erdos和Hajnal在1972年提出了确定所有$s \ge k+1$的$r_k(…
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) \…
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.
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(…
A nonconstructive Proof to show the Convergence of the $n^{th}$ root of diagonal Ramsey Number $r(n, n)$
对角Ramsey数的$n^{th}$根收敛于有限极限吗？答案是肯定的。如果一个序列满足单调性以外的收敛条件，则它可以被证明是收敛的。我们证明了这样一个属性，其中序列的$n^{th}$根不收敛，即使没有先验知识，序列是否单调或不。我们还证明了对角Ramsey数的$n^{th}$根可以表示为两个因子的乘积，第一个因子是一个已知的收敛序列，第二个因子是一个绝对收敛的无穷级数。也可以表示为其中一个乘积是收敛的，而另一个乘积的所有值来自复平面上单位圆内的一致收敛复函数全纯。我们的动机仅仅是证明猜想作为一个问题，在寻找一个解决方案，而不是建立一些深刻的理论图。第二个问题是：如果极限存在，那么极限是什么？在写这篇文章的时候，我们的理解是，所寻求的证据不需要是建设性的。在这里，我们证明了非建设性的证明，$n^{th}$根的对角拉姆齐数收敛到一个有限的限制。我们还证明了对角Ramsey数的$j^{th}$根的极限是2，其中正整数$j$依赖于Ramsey数。
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。
A linear upper bound for zero-sum Ramsey numbers of bounded degree graphs
本文研究了有限度图G和有限阿贝尔群Γ的零和Ramsey数R(G, Γ)，证明了对于任意n顶点图G和满足|Γ|整除e(G)的有限阿贝尔群Γ，R(G, Γ) ≤ Cn。
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。
A new upper bound on the minimum degree of minimal Ramsey graphs
本研究针对Burr等人1976年提出的Ramsey参数$s_r(K_k)$，证明了$s_r(K_{k+1})$的新上界为$O(k^3 r^3 \log^3 k)$，推进了极小Ramsey图最小度的相关理论研究。
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。
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。
On the geometric Ramsey numbers of trees
本文给出了树的几何Ramsey数的上界.我们证明了$R_c(T_n,H_m)=(n-1)(m-1)+1$如果$T_n$是一个毛毛虫，而$H_m$是一个关于$m$顶点的Hamilton外平面图.此外，如果$T_n$最多有两个非叶顶点，则$R_g(T_n,H_m)=(n-1)(m-1)+1$。我们还证明了$R_c(T_n,H_m)=O(n^2m)$和$R_g(T_n,H_m)=O(n^3m^2)$，如果$T_n$是$n$顶点上的任意树和$H_m$是路径宽度为2的外平面三角剖分。进一步证明了毛毛虫的几何Ramsey数的一个一致多项式上界，并给出了$R_g(T_n)$的一个上界，其中$T_n$是任意树.
A linear upper bound on the zero-sum Ramsey number of forests in $\mathbb{Z}_p$
本文研究了森林在Z_p中的零和Ramsey数，证明了当森林顶点数满足特定条件时，其零和Ramsey数不超过n+9p-12。
On degree anti-Ramsey numbers
图$H$的度反Ramsey数$AR_d(H)$是最小整数$k$，对于该最小整数，存在最大度至多为$k$的图$G$，使得$G$的任何适当的边着色产生$H$的彩虹拷贝。本文证明了度反Ramsey数的一个一般上界，确定了任意森林的度反Ramsey数的精确值，并证明了任意长度圈的度反Ramsey数的一个上界，该上界最大可能为$2$的乘法因子.我们的证明涉及各种工具，包括经典的结果Bollobás关于交叉相交的家庭和拓扑版本的霍尔定理由于Aharoni，伯杰和Meshulam。
Upper bounds for multicolour Ramsey numbers
本文给出了多重颜色Ramsey数的指数级上界，改进了Erdős和Szekeres的结果，并为r=2的情况提供了更简短的证明。
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。
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。
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