On the Turán Properties of Infinite Graphs

Let G(∞) be an infinite graph with the vertex set corresponding to the set of positive integers N. Denote by G(l) a subgraph of G(∞) which is spanned by the vertices {1, . . . , l}. As a possible extension of Turán’s theorem to infinite graphs, in this paper we will examine how large lim inf l→∞ |E(G(l))| l2 can be for an infinite graph G(∞), which does not contain an increasing path Ik with k + 1 vertices. We will show that for sufficiently large k there are Ik–free infinite graphs with 1 4 + 1 200 < lim inf l→∞ |E(G(l))| l2 . This disproves a conjecture of J. Czipszer, P. Erdős and A. Hajnal. On the other hand, we will show that lim inf l→∞ |E(G(l))| l2 ≤ 13 for any k and such G(∞).