Independence numbers and chromatic numbers of the random subgraphs of some distance graphs

This work is concerned with the Nelson-Hadwiger classical problem of finding the chromatic numbers of distance graphs in n. Most consideration is given to the class of graphs G(n, r, s)= (V(n, r), E(n, r, s)) defined as follows: where (x, y) is the Euclidean inner product. In particular, the chromatic number of G(n, 3, 1) was recently found by Balogh, Kostochka and Raigorodskii. The objects of the study are the random subgraphs (G(n, r, s), p) whose edges are independently taken from the set E(n, r, s), each with probability p. The independence number and the chromatic number of such graphs are estimated both from below and from above. In the case when r - s is a prime power and r ≤ 2s + 1, the order of growth of α((G(n, r, s), p)) and χ((G(n, r, s), p)) is established. Bibliography: 51 titles.

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