Related papers: Some recent results in Ramsey theory

This paper begins by exploring some old and new results about Ramsey numbers and minimum numbers of monochromatic triangles in $2$-colorings of complete graphs, both in the disjoint and non-disjoint cases. We then extend the theory, by…

Given a finite point set $P \subset \mathbb{R}^d$, a $k$-ary semi-algebraic relation $E$ on $P$ is the set of $k$-tuples of points in $P$, which is determined by a finite number of polynomial equations and inequalities in $kd$ real…

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…

In this paper we study the distance Ramsey number $R_{{\it D}}(s,t,d)$. The \textit{distance Ramsey number} $R_{{\it D}}(s,t,d) $ is the minimum number $n$ such that for any graph $ G $ on $ n $ vertices, either $G$ contains an induced $ s…

An ordered graph $\mathcal{G}$ is a simple graph together with a total ordering on its vertices. The (2-color) Ramsey number of $\mathcal{G}$ is the smallest integer $N$ such that every 2-coloring of the edges of the complete ordered graph…

Using computer algorithms we establish that the Ramsey number $R(3,K_{10}-e)$ is equal to 37, which solves the smallest open case for Ramsey numbers of this type. We also obtain new upper bounds for the cases of $R(3,K_k-e)$ for $11 \le k…

We prove new bounds for Ramsey numbers for book graphs $B_n$. In particular, we show that $R(B_{n-1},B_n) = 4n-1$ for an infinite family of $n$ using a block-circulant construction similar to Paley graphs. We obtain improved bounds for…

Chudnovsky, Kim, Oum, and Seymour recently established that any prime graph contains one of a short list of induced prime subgraphs [1]. In the present paper we reprove their theorem using many of the same ideas, but with the key…

Two new bounds for multicolor Ramsey numbers are proved: $R(K_3,K_3,C_4,C_4)\geq 27$ and $R_4(C_4)\leq 19$.

We show that, for every $r, k$, there is an $n = n(r,k)$ so that any $r$-coloring of the edges of the complete graph on $[n]$ will yield a monochromatic complete subgraph on vertices ${a + \sum_{i \in I} d_i \mid I \subseteq [k]}$ for some…

The study of upper density problems on Ramsey theory was initiated by Erd\H{o}s 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 $\lambda$ such that…

Ramsey theory is the study of conditions under which mathematical objects show order when partitioned. Ramsey theory on the integers concerns itself with partitions of $[1,n]$ into $r$ subsets and asks the question whether one (or more) of…

Consider the following stochastic graph process. We begin with the empty graph on n vertices and add edges one at a time, where each edge is chosen uniformly at random from the collection of potential edges that do not form triangles when…

For fixed $s \ge 3$, we prove that if optimal $K_s$-free pseudorandom graphs exist, then the Ramsey number $r(s,t) = t^{s-1+o(1)}$ as $t \rightarrow \infty$. Our method also improves the best lower bounds for $r(C_{\ell},t)$ obtained by…

A construction described by the current author (2017) uses two linear prototypes to build a compound graph with Ramsey properties inherited from the prototype graphs. The resulting graph is linear; and cyclic if both prototypes are cyclic.…

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 S\'ark\"ozy who showed that…

A numbering $f$ of a graph $G$ of order $n$ is a labeling that assigns distinct elements of the set $\left\{ 1,2,\ldots ,n\right\} $ to the vertices of $G$, where each $uv\in E\left( G\right) $ is labeled $f\left( u\right) +f\left( v\right)…

Let $F_n$ be the graph on $2n+1$ vertices consisting of $n$ triangles meeting at a single vertex. After a number of improvements over the years, it is currently known that the Ramsey number of $F_n$ is between $4.5n-5$ (Chen, Yu, Zhao) and…

We study the upper tail of the number of arithmetic progressions of a given length in a random subset of {1,...,n}, establishing exponential bounds which are best possible up to constant factors in the exponent. The proof also extends to…

In this paper we define new numbers called the Neo-Ramsay numbers. We show that these numbers are in fact equal to the Ramsay numbers. Neo-Ramsey numbers are easy to compute and for finding them it is not necessary to check all possible…