A note on multicolour Ramsey numbers and random sphere graphs
Freie Universität Berlin（柏林自由大学）；Universitat Politècnica de Catalunya (UPC)（加泰罗尼亚理工大学）
Yamaan Attwa, Albert López Vidal, Patrick Morris
备注：
4 pages. The result presented here was obtained independently by Campos and Pohoata arXiv:2601.15183 and also appears in the Master's thesis of the second author
The Ramsey number $r(t;\ell)$ is the smallest $n$ such that every $\ell$-coloring of the edges of $K_n$ gives a monochromatic $K_{t}$. In recent years, there have been several improvements on asymptotic lower bounds for these numbers when $\ell\geq 3$. This started with a breakthrough result of Conlon and Ferber, followed by further improvements of Wigderson and then Sawin. Building on the previous approaches, Sawin used blowups of an unbalanced binomial random graph to show that there is some explicit constant $δ_*\approx 0.383796$ such that $r(t;\ell)\geq 2^{δ_*(\ell-2)t+t/2+o(t)}$. In this short note, we show that one can get an exponential improvement in this bound by replacing the use of a binomial random graph with a random sphere graph, a model which which has recently been applied by Ma, Shen and Xie in a breakthrough on lower bounds for (2-colour) Ramsey numbers in the (slightly) off-diagonal setting.
Let $P_t$ denote the path on $t$ vertices. The $r$-coloured Ramsey number of $P_t$, denoted by $R_r(P_t)$, is the minimum integer $n$ such that whenever the complete graph on $n$ vertices is given an $r$-edge-colouring, there exists a monochromatic copy of $P_t$. In this note, we determine $R_r(P_5)$, which is approximately $3r$.
We prove new upper bounds on the multicolour Ramsey numbers of paths and even cycles. It is well known that $(k-1)n+o(n)\leq R_k(P_n)\leq R_k(C_n)\leq kn+o(n)$. The upper bound was recently improved by Sárközy who showed that $R_k(C_n)\leq\left(k-\frac{k}{16k^3+1}\right)n+o(n)$. Here we show $R_k(C_n) \leq (k-\frac14)n +o(n)$, obtaining the first improvement to the coefficient of the linear term by an absolute constant.
The $k$-colour bipartite Ramsey number of a bipartite graph $H$ is the least integer $N$ for which every $k$-edge-coloured complete bipartite graph $K_{N,N}$ contains a monochromatic copy of $H$. The study of bipartite Ramsey numbers was initiated over 40 years ago by Faudree and Schelp and, independently, by Gyárfás and Lehel, who determined the $2$-colour bipartite Ramsey number of paths. Recently the $3$-colour Ramsey number of paths and (even) cycles, was essentially determined as well. Improving the results of DeBiasio, Gyárfás, Krueger, Ruszinkó, and Sárközy, in this paper we determine a…
The size-Ramsey number $\hat{R}(F,r)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with $r$ colours yields a monochromatic copy of $F$. In this short note, we give an alternative proof of the recent result of Krivelevich that $\hat{R}(P_n,r) = O((\log r)r^2 n)$. This upper bound is nearly optimal, since it is also known that $\hat{R}(P_n,r) = Ω(r^2 n)$.
Given a positive integer $s$, a graph $G$ is $s$-Ramsey for a graph $H$, denoted $G\rightarrow (H)_s$, if every $s$-colouring of the edges of $G$ contains a monochromatic copy of $H$. The $s$-colour size-Ramsey number ${\hat{r}}_s(H)$ of a graph $H$ is defined to be ${\hat{r}}_s(H)=\min\{|E(G)|\colon G\rightarrow (H)_s\}$. We prove that, for all positive integers $k$ and $s$, we have ${\hat{r}}_s(P_n^k)=O(n)$, where $P_n^k$ is the $k$th power of the $n$-vertex path $P_n$.
We show that for any positive integer $r$ there exists an integer $k$ and a $k$-colouring of the edges of $K_{2^{k}+1}$ with no monochromatic odd cycle of length less than $r$. This makes progress on a problem of Erdős and Graham and answers a question of Chung. We use these colourings to give new lower bounds on the $k$-colour Ramsey number of the odd cycle and prove that, for all odd $r$ and all $k$ sufficiently large, there exists a constant $ε= ε(r) > 0$ such that $R_{k}(C_{r}) > (r-1)(2+ε)^{k-1}$.
The Multicolour Size--Ramsey Number of an Even Cycle
School of Mathematics, Nanjing University（南京大学数学系）；ECOPRO, Institute for Basic Science（基础科学研究所ECOPRO）；School of Mathematics and Statistics, Fuzhou University（福州大学数学与统计学院）
This paper sets out the results of a range of searches for linear and cyclic graph colourings with specific Ramsey properties. The new graphs comprise mainly 'template graphs' which can be used in a construction described by the current author in 2021 to build linear or cyclic compound graphs with inherited Ramsey properties. These graphs result in improved lower bounds for a wide range of multicolour Ramsey numbers. Searches were carried out using relatively simple programs (written in the language `C') to generate clauses for input to the PeneLoPe and Plingeling parallel SAT-solvers. When so…
The size-Ramsey number $\hat{R}(F)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with two colours yields a monochromatic copy of $F$. In this paper, first we focus on the size-Ramsey number of a path $P_n$ on $n$ vertices. In particular, we show that $5n/2-15/2 \le \hat{R}(P_n) \le 74n$ for $n$ sufficiently large. (The upper bound uses expansion properties of random $d$-regular graphs.) This improves the previous lower bound, $\hat{R}(P_n) \ge (1+\sqrt{2})n-O(1)$, due to Bollobás, and the upp…
A construction described by the current author in 2017 uses two linear `prototype' graphs to build a compound graph with Ramsey properties inherited from the prototypes. This paper describes a generalisation of that construction which has produced improved lower bounds in many cases for multicolour Ramsey numbers. The resulting graphs are linear, as before, and under certain specific conditions set out here, they can be cyclic. The mechanism of the new construction requires that the first prototype contains a triangle-free `template' in one colour, with defined properties. This paper shows tha…
Given graphs $H_1,H_2$, a graph $G$ is $(H_1,H_2)$-Ramsey if for every colouring of the edges of $G$ with red and blue, there is a red copy of $H_1$ or a blue copy of $H_2$. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs: this is a random graph model introduced by Bohman, Frieze and Martin in which one starts with a dense graph and then adds a given number of random edges to it. The study of Ramsey properties of randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali in 2006; they determined how many random edges must be added to a …
In this simple paper, we exhibit a Schur partition giving rise to a triangle-free linear colouring of $K_{1697}$ in 7 colours. Thus we show that the Schur number $S(7) \ge 1696$ and the multicolour Ramsey number $R_{7}(3) \ge 1698$. We also demonstrate a specific partition of the closed integer interval $[1, 376]$ which, through repeated use of a specific construction, allows us to conclude that $S(k+5) \ge 376.S(k)+160$ for any positive integer $k$. The existence of this partition, with its specific properties, implies that $\lim_{\substack{r \rightarrow \infty}} R{_r}(3)^{1/r} > 3.273$.
We prove that for any $r\in \mathbb{N}$, there exists a constant $C_r$ such that the following is true. Let $\mathcal{F}=\{F_1,F_2,\dots\}$ be an infinite sequence of bipartite graphs such that $|V(F_i)|=i$ and $Δ(F_i)\leq Δ$ hold for all $i$. Then in any $r$-edge coloured complete graph $K_n$, there is a collection of at most $\exp(C_rΔ)$ monochromatic subgraphs, each of which is isomorphic to an element of $\mathcal{F}$, whose vertex sets partition $V(K_n)$. This proves a conjecture of Corsten and Mendonça in a strong form and generalizes results on the multicolour Ramsey numbers of bounded-…
设r是一个整数，f（n）是一个函数，H是一个图.由Erdés，Hajnal，Sós和Szemerédi引入的H的r-Ramsey-Turán数RT_r（n，H，f（n））定义为n-顶点无H图G的最大边数，其中α_r（G）<= f（n），其中α_r（G）表示G的K_r-独立数.本文利用高维单位球的等周性质，构造了对任意2 <= s <= r，RT_r（n，K_{r+s}，o（n））有下界的图.这些构造对于无限族的r和s对是尖锐的。唯一以前的尖锐建设是由Bollobás和埃尔德\Hos为r = s = 2。
The anti-Ramsey number of Erdös, Simonovits and Sós from 1973 has become a classic invariant in Graph Theory. To study this invariant in Matroid Theory, we use a related invariant introduce by Arocha, Bracho and Neumann-Lara. The heterochromatic number $hc(H)$ of a non-empty hypergraph $H$ is the smallest integer $k$ such that for every colouring of the vertices of $H$ with exactly $k$ colours, there is a totally multicoloured hyperedge of $H$. Given a rank-$r$ matroid $M$, there are several hypergraphs associated to the matroid that we can consider. One is $C(M) $, the hypergraph where the po…
The first application of Szemerédi's powerful regularity method was the following celebrated Ramsey-Turán result proved by Szemerédi in 1972: any K_4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N^2 edges. Four years later, Bollobás and Erdős gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K_4-free graph on N vertices with independence number o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollobás and Erdős in 1976, several problems have been asked on estimating the minimum possible indepe…
We recursively extend the Lovász theta number to geometric hypergraphs on the unit sphere and on Euclidean space, obtaining an upper bound for the independence ratio of these hypergraphs. As an application we reprove a result in Euclidean Ramsey theory in the measurable setting, namely that every $k$-simplex is exponentially Ramsey, and we improve existing bounds for the base of the exponential.
