Mangoes and blueberries (Q1125684)

The author proves a rather innocent looking conjecture of P. Erdős and A. Hajnal. Namely, for every integer \(k\) there is a number \(f(k)\) such that if for a graph \(G\), every subgraph \(H\) of \(G\) has a stable set containing \((v(H)-k)/2\) vertices, then \(G\) contains a set \(X\) of at most \(f(k)\) vertices such that \(G \setminus X\) is bipartite. The proof is surprisingly hard and technical, and the upper bound on \(f(k)\) is huge.