Gabor Sarkozy: Monochromatic covers in edge-colored graphs and hypergraphs

We survey some results on the following problem: Say we are given fixed positive integers s, t and a family of graphs F. Minimizing over all t-edge colorings of the complete graph on n vertices, we ask for the maximum number of vertices that can be covered by at most s monochromatic members of F. This problem unites two classical problems: at one end of the spectrum (s = 1) we have the Ramsey problem, while at the other end we have cover problems. But there are some interesting problems "in-between" as well.

Several of the results are joint with András Gyárfás and/or Endre Szemerédi.