Mangoes and Blueberries

Dedicated to the memory of Paul Erdős

We prove the following conjecture of Erdős and Hajnal:

For every integer k there is an f(k) such that if for a graph G, every subgraph H of G has a stable set containing vertices, then G contains a set X of at most f(k) vertices such that GX is bipartite.

This conjecture was related to me by Paul Erdős at a conference held in Annecy during July of 1996. I regret not being able to share the answer with him.

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Received: August 20, 1997

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R, B. Mangoes and Blueberries. Combinatorica 19, 267–296 (1999).

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