Fruit Salad

Paul Erdős liked fruit salad. I mixed this one for him from ingredients obtained while working on some of his problems. He was pleased by it and carried it to several places to offer to others as well. It is very sad that I have to add to the manuscript: dedicated to his memory. 1. Nearly bipartite graphs. Szentendre, 1994. Although we are in the shade, the summer afternoon is very hot, the water Paul poured on the warm tiles of the terrace does not give too much relief. After reciting famous lines of a classical Hungarian poet (slightly rewritten by changing ‘life’ to ‘theorem’), Paul feels it is time to read from his problem book. (This has nothing to do with the BOOK, to which even he had no free access.) Soon he reads something immediately awakening my senses numbed by the humidity. ‘A problem of Hajnal and myself: assume that G is a graph in which every subset S of vertices spans a subgraph with at least b |S| 2 c − k independent vertices. Then, with some suitable function f , one can remove f(k) vertices from G so that the remaining graph is bipartite.’ It is useful to look at special cases in the process of becoming familiar with a problem. That is precisely what I am doing now, nearly two years after hearing the problem from Paul. I am looking (and working without success) at the case k = 0. Then we have a graph in which every subset of 2t vertices spans a subgraph with at least t independent vertices (the condition for the odd subsets follows from this). For easier reference, call these graphs nearly bipartite. The problem is whether nearly bipartite graphs become bipartite after the deletion of a constant number of vertices (to justify the terminology). There is an example showing that the following conjecture, if true, is best possible.