1. Noga Alon

Paul Erdős [2] conjectured in 1979 that, if in a graph on n vertices any set of [lfloor]√n[rfloor] vertices contains at least one edge, then there is a set of [lfloor]√n[rfloor] vertices that contains Ω(√n log n) edges. As observed by Erdős, this result, if true, is tight. During the workshop, and after discussions with various participants including Cameron, Erdős, Gunderson and Krivelevich, we found a proof of this conjecture, combining some probabilistic arguments with the main result of [1] (see also [3]). Hopefully this will appear in a forthcoming paper, where we also plan to include a simple proof of an extension of the main result of [1].