Distance graphs having large chromatic numbers and not containing cliques or cycles of given size

Abstract

In this work, the classical Nelson -- Hadwiger problem is studied which lies on the edge of combinatorial geometry and graph theory. It concerns colorings of distance graphs in ${\mathbb{R}}^n$, i.e., graphs such that their vertices are vectors and their edges are pairs of vectors at a distance from a given set of postive numbers apart. A series of new lower bounds are obtained for the chromatic numbers of such graphs with different restrictions on the clique numbers and the girths.

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