List coloring and Euclidean Ramsey Theory

It is well known that one can color the plane by 7 colors with no monochromatic configuration consisting of the two endpoints of a unit segment, and it is not known if a smaller number of colors suffices. Many similar problems are the subject of Euclidean Ramsey Theory, introduced by Erdős et. al. in the 70s. In sharp contrast we show that for any finite set of points K in the plane, and for any finite integer s, one can assign a list of s distinct colors to each point of the plane, so that any coloring of the plane that colors each point by a color from its list contains a monochromatic isometric copy of K. The proof follows from a general new theorem about coloring uniform simple hypergraphs with large minimum degrees from prescribed lists. 1 Euclidean Ramsey Theory A well known problem of Hadwiger and Nelson is that of determining the minimum number of colors required to color the points of the Euclidean plane so that no two points at distance 1 have the same color. Hadwiger showed already in 1945 that 7 colors suffice, and Nelson as well as L. Moser and W. Moser noted that 3 colors do not suffice. These bounds have not been improved, despite a considerable amount of effort by various researchers. A more general problem has been considered by Erdős, Graham, Montgomery, Rothschild, Spencer and Straus [4, 5, 6] under the name Euclidean Ramsey Theory. The main question is the investigation of finite point sets K in the Euclidean space for which any coloring of an Euclidean space of a sufficiently high dimension d ≥ d0(K, r) by r colors must contain a monochromatic copy of K. The conjecture is that this holds for a set K if and only if it can be embedded in a sphere. Another conjecture considered by these authors asserts that for any set K of 3 points in the plane, there is a coloring of the plane by 3 colors with no monochromatic copy of K. Intriguing variants of these questions arise when one places some restrictions on the set of colors available in each point. This is related to the notion of list coloring introduced by Vizing [8] and by Erdős, Rubin and Taylor [7].