Given a graph H, let χH (Rⁿ) be the smallest positive integer r such that there exists an r-coloring of Rⁿ with no monochromatic unit-copy of H, that is a set of |V (H) | vertices of the same color such that any two vertices corresponding to an edge of H are at distance one. This Ramsey-type function extends the famous Hadwiger--Nelson problem on the chromatic number χ (Rⁿ) =χ₊䃒 (Rⁿ) of the space from a complete graph K₂ on two vertices to an arbitrary graph H. It also extends the classical Euclidean Ramsey problem for congruent monochromatic subsets to the family of those defined by a specific subset of unit distances. Among others, we show that χH (Rⁿ) =χ (Rⁿ) for any even cycle H of length 8 or at least 12 as well as for any forest and that χH (Rⁿ) =χ (Rⁿ) /2 for any sufficiently long odd cycle. Our main tools and results, which are of independent interest, establish that Cartesian powers enjoy Ramsey-type properties for graphs with favorable Turán-type characteristics, such as zero hypercube Turán density. In addition, we prove induced variants of these results, find bounds on χH (Rⁿ) for growing dimensions n, and prove a canonical-type result. We conclude with many open problems. One of these is to determine χ₂䃔 (R²), for a cycle C₄ on four vertices.
Axenovich et al. (Wed,) studied this question.
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