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

The asymptotics of r(4,t)r(4,t)

For integers $s,t \geq 2$, the Ramsey numbers $r(s,t)$ denote the minimum $N$ such that every $N$-vertex graph contains either a clique of order $s$ or an independent set of order $t$. In this paper we prove r(4,t) = \Omega\Bigl(\frac{t^3}{\log^4 \! t}\Bigr) \quad \quad \mbox{ as }t \rightarrow \infty which determines $r(4,t)$ up to a factor of order $\log^2 \! t$, and solves a conjecture of Erd\H{o}s