An almost quadratic bound on vertex Folkman numbers

The vertex Folkman number F ( r , n , m ) , n < m , is the smallest integer t such that there exists a K m -free graph of order t with the property that every r-coloring of its vertices yields a monochromatic copy of K n . 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 ( n 2 log 4 n ) . 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.