The Zarankiewicz number z(s, m) is the maximum number of edges in a subgraph of K(s, s) that does not contain K(m, m) as a subgraph. The bipartite Ramsey number b(m, n) is the least positive integer b such that if the edges of K(b, b) are coloured with red and blue, then there always exists a blue K(m, m) or a red K(n, n). In this paper we calculate small exact values of z(s, 2) and determine b...

Let P be a graph property which is preserved by removal of edges. A random maximal P-graph is obtained from n independent vertices by randomly adding edges, at each stage choosing uniformly among edges whose inclusion would not destroy property P, until no further edges can be added. We address the question of the number of edges of a random maximal P-graph for several properties P, in particul...

A coloring of a complete bipartite graph is shuffle-preserved if it is the case that assigning a color c to edges (u, v) and (u′, v′) enforces the same color assignment for edges (u, v′) and (u′, v). (In words, the induced subgraph with respect to color c is complete.) In this paper, we investigate a variant of the Ramsey problem for the class of complete bipartite multigraphs. (By a multigraph...

Let G be a subgraph of a complete bipartite graph Kn,n. Let N(G) be a 0-1 incidence matrix with edges of Kn,n against images of G under the automorphism group of Kn,n. A diagonal form of N(G) is found for every G, and whether the row space of N(G) over Zp contains the vector of all 1’s is determined. This re-proves Caro and Yuster’s results on zero-sum bipartite Ramsey numbers [3], and provides...

• We obtain the upper bounds for number of edges in even-cycle-free subgraphs complete transposition graphs, which are a family bipartite and arctransitive graphs. A new auxiliary graphs related to constructed used order bounds. Ramsey-type result is obtained. Given G H , generalized Turán ex ( ) maximum an -free subgraph . In this paper, we asymptotic bound on C T n 2 ℓ any ≥ 3 where cycle len...

A pair of graphs (Hb,Hr) is highly Ramsey-infinite if there is some constant c such that for large enough n there are at least 2cn 2 non-isomorphic graphs on n or fewer vertices that are minimal with respect to the property that when their edges are coloured blue or red, there is necessarily a blue copy of Hb or a red copy of Hr. We show that a pair of 3-connected graphs is highly Ramsey-infini...