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

Chromatic Vertex Folkman Numbers

For graph G and integers a1 ≥ · · · ≥ ar ≥ 2, we write G → (a1, · · · , ar) if and only if for every r-coloring of the vertex set V (G) there exists a monochromatic Kai in G for some color i ∈ {1, · · · , r}. The vertex Folkman number Fv(a1, · · · , ar; s) is defined as the smallest integer n for which there exists a Ks-free graph G of order n such that G→ (a1, · · · , ar). It is well known tha...

Ks-Free Graphs Without Large Kr-Free Subgraphs

Large Kr-free subgraphs in Ks-free graphs and some other Ramsey-type problems

In this paper we present three Ramsey-type results, which we derive from a simple and yet powerful lemma, proved using probabilistic arguments. Let 3 ≤ r < s be fixed integers and let G be a graph on n vertices not containing a complete graph Ks on s vertices. More than 40 years ago Erdős and Rogers posed the problem of estimating the maximum size of a subset of G without a copy of the complete...

An Ordered Turán Problem for Bipartite Graphs

Let F be a graph. A graph G is F -free if it does not contain F as a subgraph. The Turán number of F , written ex(n, F ), is the maximum number of edges in an F -free graph with n vertices. The determination of Turán numbers of bipartite graphs is a challenging and widely investigated problem. In this paper we introduce an ordered version of the Turán problem for bipartite graphs. Let G be a gr...

C4-saturated bipartite graphs

Let H be a graph. A graph G is said to be H -free if it contains no subgraph isomorphic to H . A graph G is said to be an H -saturated subgraph of a graph K if G is an H -free subgraph of K with the property that for any edge e∈E(K) \ E(G); G ∪ {e} is not H -free. We present some general results on Ks; t-saturated subgraphs of the complete bipartite graph Km;n and study the problem of 3nding, f...