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 Multicolor Ramsey Numbers Involving Cycles

Establishing the values of Ramsey numbers is, in general, a difficult task from the computational point of view. Over the years, researchers have developed methods to tackle this problem exhaustively in ways that require intensive computations. These methods are often backed by theoretical results that allow us to cut the search space down to a size that is within the limits of current computing capacity. This thesis focuses on developing algorithms and applying them to generate Ramsey colorings avoiding cycles. It adds to a recent trend of interest in this particular area of finite Ramsey theory. Our main contributions are the enumeration of all (C_5,C_5,C_5;n) Ramsey colorings and the study of the Ramsey numbers R(C_4,C_4,K_4) and R4(C_5)

Semi-algebraic Ramsey numbers

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 variables. The description complexity of such a relation is at most $t$ if the number of polynomials and their degrees are all bounded by $t$. The Ramsey number $R^{d,t}_k(s,n)$ is the minimum $N$ such that any $N$-element point set $P$ in $\mathbb{R}^d$ equipped with a $k$-ary semi-algebraic relation $E$, such that $E$ has complexity at most $t$, contains $s$ members such that every $k$-tuple induced by them is in $E$, or $n$ members such that every $k$-tuple induced by them is not in $E$. We give a new upper bound for $R^{d,t}_k(s,n)$ for $k\geq 3$ and $s$ fixed. In particular, we show that for fixed integers $d,t,s$, $R^{d,t}_3(s,n) \leq 2^{n^{o(1)}},$ establishing a subexponential upper bound on $R^{d,t}_3(s,n)$. This improves the previous bound of $2^{n^C}$ due to Conlon, Fox, Pach, Sudakov, and Suk, where $C$ is a very large constant depending on $d,t,$ and $s$. As an application, we give new estimates for a recently studied Ramsey-type problem on hyperplane arrangements in $\mathbb{R}^d$. We also study multi-color Ramsey numbers for triangles in our semi-algebraic setting, achieving some partial results

Bipartite Ramsey Numbers and Zarankiewicz Numbers

The bipartite Ramsey number b(m, n) is the minimum b such that any 2-coloring of Kb,b results in a monochromatic Km,m subgraph in the first color or a monochromatic Kn,n subgraph in the second color. The Zarankiewicz number z(m, n; s, t) is the maximum size among Ks,t-free subgraphs of Km,n. In this work, we discuss the intimate relationship between the two numbers as well as propose a new method for bounding the Zarankiewicz numbers. We use the better bounds to improve the upper bound on b(2, 5), in addition we improve the lower bound of b(2, 5) by construction. The new bounds are shown to be 17 ≤ b(2, 5) ≤ 18. Additionally, we apply the same methods to the multicolor case b(2, 2, 3) which has previously not been studied and determine bounds to be 16 ≤ b(2, 2, 3) ≤ 23

Ramsey numbers for set-colorings

For s, t, n ∈ N with s ≥ t, an (s, t)-coloring of K$_n$ is an edge coloring of Kn in which each edge is assigned a set of t colors from {1, . . . , s}. For k ∈ N, a monochromatic K$_k$ is a set of k vertices S such that for some color i ∈ [s], i ∈ c(uv) for all distinct u, v ∈ S. As in the case of the classical Ramsey number, we are interested in the least positive integer n = R$_{s,t}$(k) such that for any (s, t)-coloring of K$_n$, there exists a monochromatic K$_k$. We estimate upper and lower bounds for general cases and calculate close bounds for some small cases of R$_{s,t}$(k)