For fixed integersp, q an edge coloring of a complete graphK is called a (p, q)-coloring if the edges of everyKp⊑K are colored with at leastq distinct colors. Clearly, (p, 2)-colorings are the classical Ramsey colorings without monochromaticKp subgraphs. Letf(n, p, q) be the minimum number of colors needed for a (p, q)-coloring ofKn. We use the Local Lemma to give a general upper bound forf. We determine for everyp the smallestq for whichf(n, p, q) is linear inn and the smallestq for whichf(n, p, q) is quadratic inn. We show that certain special cases of the problem closely relate to Turán type hypergraph problems introduced by Brown, Erdős and T. Sós. Other cases lead to problems concerning proper edge colorings of complete graphs.
Mathematics Subject Classification (1991)05D10
Unable to display preview. Download preview PDF.
- W. G. Brown, P. Erdős, V. T. Sós: Some extremal problems onr-graphs, inNew directions in the theory of Graphs, Proc. 3rd Ann Arbor Conference on Graph Theory, 53–63, Academic Press, New York, 1973.Google Scholar
- W. G. Brown, P. Erdős, V. T. Sós: On the existence of triangulated spheres in 3-graphs and related problems,Periodica Mathematica Hungarica,3 (1973), 221–228.Google Scholar
- P. Erdős: Solved and unsolved problems in combinatorics and combinatorial number theory,Congressus Numerantium,32 (1981), 49–62.Google Scholar
- P. Erdős: Extremal problems in graph theory, inTheory of Graphs and its Applications (M. Fiedler, ed.), Academic Press, New York, 1964, 29–36.Google Scholar
- P. Erdős, D. J. Kleitman: On coloring graphs to maximize the proportion of multicoloredk-edges,J. of Combinatorial Theory,5 (1968), 164–169.Google Scholar
- P. Erdős, L. Lovász: Problems and results on 3-chromatic hypergraphs and some related problems, in:Infinite and Finite sets, Colloquia Math. Soc. J. Bolyai vol. 10 (A. Hajnal et al eds.) North-Holland, Amsterdam, 609–617 1995.Google Scholar
- P. Erdős, V. T. Sós: personal communication, 1994.Google Scholar
- P. C. Fishburn: personal communication, 1995.Google Scholar
- A. Gyárfás, J. Lehel: Linear Sets with Five Distinct Differences among any Four Elements,J. of Combinatorial Theory B,64 (1995), 108–118.Google Scholar
- I. Z. Ruzsa, E. Szemerédi: Triple systems with no six points carrying three triangles, inCombinatorics (Keszthely, 1976), Coll. Math. Soc. J. Bolyai 18, Volume II. 939–945.Google Scholar
- W. D. Wallis: Combinatorial Designs, Monographs and Textbooks inPure and Applied Mathematics, Vol. 118, Marcel Dekker, Inc. 1988.Google Scholar