On K s -free subgraphs in K s+k -free graphs and vertex Folkman numbers

Abstract

Extending the problem of determining Ramsey numbers Erdős and Rogers introduced the following function. For given integers 2 ≤ s < t let fs,t(n) = min˘max{|S| : S ⊆ V (H) and H [S] contains no Ks}¯, where the minimum is taken over all Kt-free graphs H of order n. This function attracted a considerable amount of attention but despite that, the gap between the lower and upper bounds is still fairly wide. For example, when t = s+1, the best bounds have been of the form Ω`n 1 2 +o(1)´≤ fs,s+1(n) ≤ O(n 1− (s) ), where (s) tends to zero as s tends to infinity. In this paper we improve the upper bound by showing that fs,s+1(n) ≤ O`n 2 3´. Moreover, we show that for every ε > 0 and sufficiently large integers 1 k s, Ω`n 1 2 −ε´≤ f s,s+k (n) ≤ O`n 1 2 +ε´. In addition, we also discuss some connections between the function fs,t and vertex Folkman numbers.

Improved bounds for the Erdős-Rogers function

“…Of course, this on its own does nothing, since a complete tripartite graph has a huge independent set, but we can use it as a building block by taking a union of many complete tripartite graphs. In previous constructions, such as Wolfovitz's graph that gives an upper bound for f 3,4 (n), the vertex sets of these tripartite graphs are chosen algebraically -in Wolfovitz's case they are the lines of a projective plane. The main difference in our approach is that we simply choose them at random, where the number we choose and the size of each one are parameters that we optimize at the end of the argument.…”

On the Nonexistence of Some Generalized Folkman Numbers

“…The best known concrete lower and upper bounds on various Ramsey numbers of the form R(J s , K t ) are collected in [14]; for example, we know that 30 ≤ R(J 5 , K 5 ) ≤ 33. In that case, any 29-vertex witness graph to Ramsey lower bound seems to be a good candidate for the vertex Folkman number case of arrowing (3,4) v . This would give an interesting bound F v (K 3 , K 4 ; J 5 ) ≤ 29 (unfortunately, we were not successful in finding any such graph so far).…”

“…The cases when H 1 , H 2 and H are complete graphs have been studied by many authors, for two and more colors, in particular in [1,2,3,4,5,6,9,10,11,12,13,15,17]. Often, if the graphs H i and H are complete, we will simply write the order of the graph, say, as in F e (s, t; k) instead of F e (K s , K t ; K k ).…”

On the Ramsey–Turán number with small s-independence number

“…Krivelevich [1], Dudek and Rödl [13], Wolfovitz [36], and most recently by Dudek and Mubayi [12], and Dudek, Rödl and Retter [14]. Due to [12] and [14] it is known that for any s ≥ 3, Ω n log n log log n = f s,s+1 (n) = O (log n) 4s 2 √ n .…”