Constructive Lower Bounds on Classical Multicolor Ramsey Numbers

This paper studies lower bounds for classical multicolor Ramsey numbers, first by giving a short overview of past results, and then by presenting several general constructions establishing new lower bounds for many diagonal and off-diagonal multicolor Ramsey numbers. In particular, we improve several lower bounds for R_k(4) and R_k(5) for some small k, including 415 \u3c = R_3(5), 634 \u3c = R_4(4), 2721 \u3c = R_4(5), 3416 \u3c = R_5(4) and 26082 \u3c = R_5(5). Most of the new lower bounds are consequences of general constructions

Small Ramsey Numbers

We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge. Many results pertaining to other more studied cases are also presented. We give references to all cited bounds and values, as well as to previous similar compilations. We do not attempt complete coverage of asymptotic behavior of Ramsey numbers, but concentrate on their specific values

On a Diagonal Conjecturefor Classical Ramsey Numbers

Let R(k1, · · · , kr) denote the classical r-color Ramsey number for integers ki ≥ 2. The Diagonal Conjecture (DC) for classical Ramsey numbers poses that if k1, · · · , kr are integers no smaller than 3 and kr−1 ≤ kr, then R(k1, · · · , kr−2, kr−1 − 1, kr + 1) ≤ R(k1, · · · , kr). We obtain some implications of this conjecture, present evidence for its validity, and discuss related problems. Let Rr(k) stand for the r-color Ramsey number R(k, · · · , k). It is known that limr→∞ Rr(3)1/r exists, either finite or infinite, the latter conjectured by Erd˝os. This limit is related to the Shannon capacity of complements of K3-free graphs. We prove that if DC holds, and limr→∞ Rr(3)1/r is finite, then limr→∞ Rr(k) 1/r is finite for every integer k ≥ 3

THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References

and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a

Some Computational and Theoretical Problems for Ramsey Numbers

We discuss some computational challenges and related open questions concerning classical Ramsey numbers. This talk overviews known constructive bounds for the difference between consecutive Ramsey numbers and presents what is known about the most studied cases including $R(5,5)$ and $R(3,3,4)$. Although the main problems we discuss are concerned with concrete cases, and they involve significant computational approaches, there are interesting and important theoretical questions behind each of them

Semi-algebraic colorings of complete graphs

We consider $m$-colorings of the edges of a complete graph, where each color class is defined semi-algebraically with bounded complexity. The case $m = 2$ was first studied by Alon et al., who applied this framework to obtain surprisingly strong Ramsey-type results for intersection graphs of geometric objects and for other graphs arising in computational geometry. Considering larger values of $m$ is relevant, e.g., to problems concerning the number of distinct distances determined by a point set. For $p\ge 3$ and $m\ge 2$, the classical Ramsey number $R(p;m)$ is the smallest positive integer $n$ such that any $m$-coloring of the edges of $K_n$, the complete graph on $n$ vertices, contains a monochromatic $K_p$. It is a longstanding open problem that goes back to Schur (1916) to decide whether $R(p;m)=2^{O(m)}$, for a fixed $p$. We prove that this is true if each color class is defined semi-algebraically with bounded complexity. The order of magnitude of this bound is tight. Our proof is based on the Cutting Lemma of Chazelle {\em et al.}, and on a Szemer\'edi-type regularity lemma for multicolored semi-algebraic graphs, which is of independent interest. The same technique is used to address the semi-algebraic variant of a more general Ramsey-type problem of Erd\H{o}s and Shelah

An Improvement to Mathon’s Cyclotomic Ramsey Colorings

In this note we show how to extend Mathon’s cyclotomic colorings of the edges of some complete graphs without increasing the maximum order of monochromatic complete subgraphs. This improves the well known lower bound construction for multicolor Ramsey numbers, in particular we obtain R3(7) ≥ 3214