A ramsey type problem concerning vertex colourings

On a Ramsey-type problem

On a Ramsey Type Problem of Shelah

Shelah used a Ramsey type statement in his new proof of Vander Waerden's theorem ([5]). It was formulated in [3] as a coloring problem on mesh clique graphs: find the smallest n = F(r) such that in any r-coloring of the edges of Kn x Kn there exists a K 2 x K2 whose opposite edges have the same color. Here the trivial upper bound on F(r) is improved using the Fisher inequality on hypergraphs wi...

Constructive Bounds for a Ramsey-Type Problem

For every fixed integers r, s satisfying 2 ≤ r < s there exists some = (r, s) > 0 for which we construct explicitly an infinite family of graphs Hr,s,n, where Hr,s,n has n vertices, contains no clique on s vertices and every subset of at least n1− of its vertices contains a clique of size r. The constructions are based on spectral and geometric techniques, some properties of Finite Geometries a...

On a Ramsey-Turán type problem

Denote by I(G) the maximal number of independent points in a graph G and let or(G) = I(G), where G is the complement of G. Thus a(G) is the maximal p for which G contains a K, , a complete graph with p vertices. Denote byf(n, k, 1) the maximal m for which there is a graph with n points and m edges such that or(G) < k and I(G) < I. The function f(n, k, I) was introduced and investigated by Erdii...

Avoidance colourings for small nonclassical Ramsey numbers

The irredundant Ramsey number s = s(m,n) [upper domination Ramsey number u = u(m,n), respectively] is the smallest natural number s [u, respectively] such that in any red-blue edge colouring (R,B) of the complete graph of order s [u, respectively], it holds that IR(B) ≥ m or IR(R) ≥ n [Γ(B) ≥ m or Γ(R) ≥ n, respectively], where Γ and IR denote respectively the upper domination number and the ir...