A Folkman Linear Family

For graphs $F$ and $G$, let $F\to (G,G)$ signify that any red/blue edge coloring of $F$ contains a monochromatic $G$. Define Folkman number $f(G;p)$ to be the smallest order of a graph $F$ such that $F\to (G,G)$ and $\omega(F) \le p$. It is shown that $f(G;p)\le cn$ for graphs $G$ of order $n$ with $\Delta(G)\le \Delta$, where $\Delta\ge 3$, $c=c(\Delta)$ and $p=p(\Delta)$ are positive constants.Comment: 11 page

Combinatorial theorems relative to a random set

We describe recent advances in the study of random analogues of combinatorial theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201

On globally sparse Ramsey graphs

We say that a graph $G$ has the Ramsey property w.r.t.\ some graph $F$ and some integer $r\geq 2$, or $G$ is $(F,r)$-Ramsey for short, if any $r$-coloring of the edges of $G$ contains a monochromatic copy of $F$. R{\"o}dl and Ruci{\'n}ski asked how globally sparse $(F,r)$-Ramsey graphs $G$ can possibly be, where the density of $G$ is measured by the subgraph $H\subseteq G$ with the highest average degree. So far, this so-called Ramsey density is known only for cliques and some trivial graphs $F$. In this work we determine the Ramsey density up to some small error terms for several cases when $F$ is a complete bipartite graph, a cycle or a path, and $r\geq 2$ colors are available

Algorithms for bounding Folkman numbers

For an undirected, simple graph G, we write G -\u3e (a_1,...,a_k)^v (G -\u3e (a_1,...,a_k)^e) if for every vertex (edge) k-coloring, a monochromatic K_(a_i) is forced in some color i in {1,...,k}. The vertex (edge) Folkman number is defined as F_v(a_1,...,a_k;p) = min{|V(G)| : G -\u3e (a_1,...,a_k;p)^v, K_p not in G} F_e(a_1,...,a_k;p) = min{|V(G)| : G -\u3e (a_1,...,a_k;p)^e, K_p not in G} for p \u3e max{a_1,...,a_k}. Folkman showed in 1970 that these numbers always exist for valid values of p. This thesis concerns the computation of a new result in Folkman number theory, namely that F_v(2,2,3;4)=14. Previously, the bounds stood at 10 \u3c= F_v(2,2,3;4) \u3c= 14, proven by Nenov in 2000. To achieve this new result, specialized algorithms were executed on the computers of the Computer Science network in a distributed processing effort. We discuss the mathematics and algorithms used in the computation. We also discuss ongoing research into the computation of the value of F_e(3,3;4). The current bounds stand at 16 \u3c= F_e(3,3;4) \u3c= 3e10^9. This number was once the subject of an Erd s prize---claimed by Spencer in 1988

An almost quadratic bound on vertex Folkman numbers

AbstractThe vertex Folkman number F(r,n,m), n<m, is the smallest integer t such that there exists a Km-free graph of order t with the property that every r-coloring of its vertices yields a monochromatic copy of Kn. The problem of bounding the Folkman numbers has been studied by several authors. However, in the most restrictive case, when m=n+1, no polynomial bound has been known for such numbers. In this paper we show that the vertex Folkman numbers F(r,n,n+1) are bounded from above by O(n2log4n). Furthermore, for any fixed r and any small ε>0 we derive the linear upper bound when the cliques bigger than (2+ε)n are forbidden