The Sub-Exponential Transition for the Chromatic Generalized Ramsey Numbers
A simple graph-product type construction shows that for all natural numbers r≥q, there exists an edge-coloring of the complete graph on 2 r vertices using r colors where the graph consisting of the union of any q color classes has chromatic number 2 q . We show that for each fixed natural number q, if there exists an edge-coloring of the complete graph on n vertices using r colors where the graph consisting of the union of any q color classes has chromatic number at most 2 q − 1, then n must be sub-exponential in r. This answers a question of Conlon, Fox, Lee, and Sudakov.
Mathematics Subject Classification (2010)05C55 05D10
Unable to display preview. Download preview PDF.
- D. Conlon, J. Fox, C. Lee and B. Sudakov: On the grid Ramsey problem and related questions, Int. Math. Res. Not..Google Scholar
- D. Conlon, J. Fox, C. Lee and B. Sudakov: The Erdős-Gyárfás problem on generalized Ramsey numbers, Proc. London Math. Soc., to appear.Google Scholar
- P. Erdős: Problems and results on finite and infinite graphs, in: Recent advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974), 183–192, Academia, Prague, 1975.Google Scholar
- Y. Peng, V. Rödl and A. Ruciński: Holes in graphs, Electron. J. Combin. 9 (2002), Research Papers 1.Google Scholar