A connected graph is double connected if its complement is also connected The following Ramsey type theorem is proved in this paper There exists a function h n de ned on the set of integers exceeding three such that every double connected graph on at least h n vertices must contain as an induced subgraph a special double connected graph which is either one of the following graphs or the complem...

Suppose .F is a collection of 3-subsets of (1,2,..., n). The problem of determining the least integer f(n, k) with the property that if l,FT/ >f(n, k) then .7 contains a k-star (i.e., k 3-sets such that the intersection of any pair of them consists of exactly the same element) is studied. It is proved that, for k odd, f(n, k) = k(k 1) n + P(k-‘) and, for k even,f(n, k) = k(k 3/2) n + F(n + k’).

Abstract Which patterns must a two-colouring of $K_n$ contain if each vertex has at least $\varepsilon n$ red and blue neighbours? We show that when \gt 1/4$ , complete subgraph on $\Omega (\log n)$ vertices where one the colours forms balanced bipartite graph. When \leq this statement is no longer true, as evidenced by following colouring $\chi$ . Divide set into $4$ parts nearly equal in size...