论文信息 - On a Frankl-Rödl theorem and its geometric corollaries

On a Frankl-Rödl theorem and its geometric corollaries

Abstract There are two main results in this paper. First, for any d ∈ N , we find an explicit c d > 1 such that χ d ( R n ) ≥ ( c d + o ( 1 ) ) n , where χ d ( R n ) is the minimum number of colors needed to color all points in R n so that there is no monochromatic set of vertices of a d-dimensional regular simplex. Second, for any k ≥ 3 we find an explicit ξ k > 1 such that for any n ∈ N , there exists a distance graph in R n with girth at least k and the chromatic number at least ( ξ k + o ( 1 ) ) n .

A. A. Sagdeev | A. Sagdeev

[1] Andrei Raigorodskii,et al. Distance graphs having large chromatic numbers and containing no cliques or cycles of a given size , 2013 .

[2] A. Raigorodskii,et al. Improvements of the Frankl-Rödl theorem on the number of edges of a hypergraph with forbidden intersections, and their consequences in the problem of finding the chromatic number of a space with forbidden equilateral triangle , 2015 .

[3] Andrey Kupavskii. Distance graphs with large chromatic number and arbitrary girth , 2013, ArXiv.

[4] V. Rödl,et al. A partition property of simplices in Euclidean space , 1990 .

[5] I. Kríz. Permutation groups in Euclidean Ramsey theory , 1991 .

[6] V. Rödl,et al. All triangles are Ramsey , 1986 .

[7] Kristal Cantwell. All regular polytopes are Ramsey , 2007, J. Comb. Theory, Ser. A.

[8] J. Spencer. Ramsey Theory , 1990 .

[9] A. Raigorodskii,et al. On the chromatic number of a space with forbidden equilateral triangle , 2014 .

[10] C. Rogers,et al. The realization of distances within sets in Euclidean space , 1972 .

[11] Improvement of the Frankl-Rödl theorem on the number of edges in hypergraphs with forbidden cardinalities of edge intersections , 2014 .

[12] Andrei Mikhailovich Raigorodskii. On the chromatic number of a space , 2000 .

[13] P. Erdös,et al. Graph Theory and Probability , 1959 .