Polychromatic Coloring of Tuples in Hypergraphs

A hypergraph H consists of a set V of vertices and a set E of hyperedges that are subsets of V. A t-tuple of H is a subset of t vertices of V. A t-tuple k-coloring of H is a mapping of its t-tuples into k colors. A coloring is called (t, k, f)-polychromatic if each hyperedge of E that has at least f vertices contains tuples of all the k colors. Let fH(t, k) be the minimum f such that H has a (t, k, f)-polychromatic coloring. For a family of hypergraphs H let fH(t, k) be the maximum fH(t, k) over all hypergraphs H in H. Determining fH(t, k) has been an active research direction in recent years. This is challenging even for t = 1. We present several new results in this direction for t ≥ 2. Let H be the family of hypergraphs H that is obtained by taking any set P of points in R2, setting V := P and E := {d ∩ P : d is a disk in R2}. We prove that fH(2, k) ≤ 3.7k, that is, the pairs of points (2-tuples) can be k-colored such that any disk containing at least 3.7k points has pairs of all colors. We generalize this result to points and balls in higher dimensions. For the family H of hypergraphs that are defined by grid vertices and axis-parallel rectangles in the plane, we show that fH(2, k) ≤ √ck ln k for some constant c. We then generalize this to higher dimensions, to other shapes, and to tuples of larger size. For the family H of shrinkable hypergraphs of VC-dimension at most d we prove that fH(d+1, k) ≤ ck for some constant c = c(d). Towards this bound, we obtain a result of independent interest: Every hypergraph with n vertices and with VC-dimension at most d has a (d+1)-tuple T of depth at least n/c, i.e., any hyperedge that contains T also contains n/c other vertices. For the relationship between t-tuple coloring and vertex coloring in any hypergraph H we establish the inequality 1/e · tk1/t ≤ fH(t, k) ≤ fH(1, tk1/t). For the special case of k = 2, referred to as the bichromatic coloring, we prove that t + 1 ≤ fH(t, 2) ≤ max{fH(1, 2), t + 1}; this improves upon the previous best known upper bound. We study the relationship between tuple coloring and epsilon nets. In particular we show that if fH(1, k) = O(k) for a hypergraph H with n vertices, then for any 0 < ∈ < 1 the t-tuples of H can be partitioned into Ω ((∈n/t)t) ∈-t-nets. This bound is tight when t is a constant.