A. A. Sagdeev

TL;DR: An exponentially growing lower bound on the chromatic number of distance graphs with large girth is improved by improving known upper bounds on the product of cardinalities of two families of homogeneous subsets with one forbidden cross-intersection.

Abstract: We find a new analogue of the Frankl-Wilson theorem on the independence number of distance graphs of some special type. We apply this new result to two combinatorial geometry problems. First, we improve a previously known value c such that χ (R;S2) ≥ (c+ o(1)), where χ (R;S2) is the minimum number of colors needed to color all points of Rn so that there is no monochromatic set of vertices of a unit equilateral triangle S2. Second, given m ≥ 3 we improve the value ξm such that for any n ∈ N there is a distance graph in Rn with the girth greater than m and the chromatic number at least (ξm + o(1))n.

TL;DR: For some sets, for the first time, explicit exponentially growing lower bounds are obtained for the corresponding chromatic numbers of spaces with forbidden monochromatic sets.

TL;DR: The Frankl-Rodl classical bound for the number of edges in a hypergraph with forbidden intersections was improved in this paper, and the improvements were used to obtain new results in Euclidean Ramsey theory and in combinatorial geometry.