How Tight Is the Bollobás-Komlós Conjecture?

Abstract.

The bipartite case of the Bollobás and Komlós conjecture states that for every Δ₀, γ>0 there is an α=α(Δ₀, γ) >0 such that the following statement holds: If G is any graph with minimum degree at least then G contains as subgraphs all n vertex bipartite graphs, H, satisfying¶

¶Here b(H), the bandwidth of H, is the smallest b such that the vertices of H can be ordered as v ₁, …, v _n such that v _i∼_H v _j implies |i−j|≤b.¶ This conjecture has been proved in [1]. Answering a question of E. Szemerédi [6] we show that this conjecture is tight in the sense that as γ→0 then α→0. More precisely, we show that for any there is a Δ₀ such that that α(Δ₀, γ)≤4 γ.