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On the Bollobás-Eldrige conjecture for bipartite graphs. (English) Zbl 1136.05029
The following result, a special case of the notorious Bollobás-Eldridge conjecture, is proved. If ${{\Delta }}_{1}\ge {{\Delta }}_{2}\ge 2$ are integers, then there exist a natural number ${n}_{0}$ and a real $\beta <{{\Delta }}_{2}/\left({{\Delta }}_{2}+1\right)$ such that if $n\ge {n}_{0}$, $H$ and $G$ are graphs on $n$ vertices, $H$ is bipartite with maximal degrees ${{\Delta }}_{1}$, ${{\Delta }}_{2}$ on the respective sides, $G$ has minimal degree $>\beta n$, then $H$ can be placed into $G$. The involved proof uses the Regularity Lemma and a variant (formulated and proved by the author) of the Blow Up Lemma of J. Komlós, G. N. Sárközy, and E. Szemerédi.
##### MSC:
 05C35 Extremal problems (graph theory)