Infinite Turán problems for bipartite graphs

الأصل · EN

We consider an infinite version of the bipartite Turán problem. Let G be an infinite graph with V(G) = N and let Gₙ be the n-vertex subgraph of G induced by the vertices {1,2,, n }. We show that if G is K₂,ₜ₊₁-free then for infinitely many n, e(Gₙ) ≤ 0.471 √t n³/². Using the K₂,ₜ₊₁-free graphs constructed by Füredi, we construct an infinite K₂,ₜ₊₁-free graph with e(Gₙ) ≥ 0.23 √tn³/² for all n ≥ n₀.