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A variant of the classical Ramsey problem. (English) Zbl 0910.05034
The following quantity is estimated. Let $$f(n,p,q)$$ be the minimum number of colors needed to color all edges of $$K_n$$ such that every $$K_p$$ gets at least $$q$$ colors. A general upper bound is given using the Lovász local lemma. If $$q={p\choose 2}-p+3$$ then $$f(n,p,q)$$ is linear while $$f(n,p,q-1)$$ is sublinear. If $$q={p\choose 2}-\lfloor{p\over 2}\rfloor+2$$ then $$f(n,p,q)=\Omega(n^2)$$ while $$f(n,p,q-1)=O(n^{2-{4\over p}})$$ but is $$\Omega(n^{{4\over 3}})$$ for $$p\geq 7$$. $$f(n,p,p)=\Omega(n^{{1\over{p-2}}})$$. Also, $${5\over 6}(n-1)\leq f(n,4,5)$$ and $$f(n,9,34)={n\choose 2}-o(n^2)$$.

##### MSC:
 05C35 Extremal problems in graph theory 05C80 Random graphs (graph-theoretic aspects) 05D10 Ramsey theory 05C55 Generalized Ramsey theory
##### Keywords:
extremal graph theory; probabilistic methods
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##### References:
 [1] W. G. Brown, P. Erdos, 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. [2] W. G. Brown, P. Erdos, V. T. Sós: On the existence of triangulated spheres in 3-graphs and related problems,Periodica Mathematica Hungarica,3 (1973), 221–228. · Zbl 0269.05111 · doi:10.1007/BF02018585 [3] P. Erdos: Solved and unsolved problems in combinatorics and combinatorial number theory,Congressus Numerantium,32 (1981), 49–62. [4] P. Erdos: Extremal problems in graph theory, inTheory of Graphs and its Applications (M. Fiedler, ed.), Academic Press, New York, 1964, 29–36. [5] P. Erdos, D. J. Kleitman: On coloring graphs to maximize the proportion of multicoloredk-edges,J. of Combinatorial Theory,5 (1968), 164–169. · Zbl 0167.22302 · doi:10.1016/S0021-9800(68)80051-1 [6] P. Erdos, 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. [7] P. Erdos, V. T. Sós: personal communication, 1994. [8] P. C. Fishburn: personal communication, 1995. [9] A. Gyárfás, J. Lehel: Linear Sets with Five Distinct Differences among any Four Elements,J. of Combinatorial Theory B,64 (1995), 108–118. · Zbl 0830.05061 · doi:10.1006/jctb.1995.1028 [10] 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. [11] W. D. Wallis: Combinatorial Designs, Monographs and Textbooks inPure and Applied Mathematics, Vol. 118, Marcel Dekker, Inc. 1988.
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