Semi-algebraic colorings of complete graphs

In this paper, we consider edge colorings of the complete graph, where the vertices are points in R, and each color class Ei is defined by a semi-algebraic relation of constant complexity on the point set. One of our main results is a multicolor regularity lemma: For any 0 < ε < 1, the vertex set of any such edge colored complete graph with m colors can be equitably partitioned into at most (m/ε) parts, such that all but at most an ε-fraction of the pairs of parts are monochromatic between them. Here c > 0 is a constant that depends on the dimension d and the complexity of the semi-algebraic relations. This generalizes a theorem of Alon, Pach, Pinchasi, Radoičić and Sharir, and Fox, Pach, and Suk. As an application, we prove the following result on generalized Ramsey numbers for such semi-algebraic edge colorings. For fixed integers p and q with 2 ≤ q ≤ ( p 2 ) , a (p, q)-coloring is an edge-coloring of a complete graph in which every p vertices induce at least q distinct colors. The function f∗(n, p, q) is the minimum integer m such that there is a (p, q)-coloring of Kn with at most m colors, where the vertices of Kn are points in R, and each color class can be defined as a semi-algebraic relation of constant complexity. Here we show that f∗(n, p, dlog pe + 1) ≥ Ω ( n 1 c log2 p ) , and f∗(n, p, dlog pe) ≤ O(log n), thus determining the exact value of q at which f∗(n, p, q) changes from logarithmic to polynomial in n. We also show the following result on a distinct distances problem of Erdős. Let V be an n-element planar point set such that any p members of V determine at least ( p 2 ) − p+ 6 distinct distances. Then V determines at least n 8 7−o(1) distinct distances.