Let HD d (p, q) denote the minimal size of a transversal that can always be guaranteed for a family of compact convex sets in R d which satisfy the (p, q)-property (p ≥ q ≥ d + 1). In a celebrated proof of the Hadwiger-Debrunner conjecture, Alon and Kleitman proved thatThis paper has two parts. In the first part we present several improved bounds on HD d (p, q). In particular, we obtain the first near tight estimate of HD d (p, q) for an extended range of values of (p, q) since the 1957 Hadwiger-Debrunner theorem.In the second part we prove a (p, 2)-theorem for families in R 2 with union complexity below a specific quadratic bound. Based on this, we introduce a polynomial time constant factor approximation algorithm for MAX-CLIQUE of intersection graphs of convex sets satisfying this property. It is not likely that our constant factor approximation can be improved to a PTAS as MAX-CLIQUE for intersection graphs of fat ellipses is known to be APX-HARD and fat ellipses have sub-quadratic union complexity.

In this paper we present a complete characterization of the smallest sets which block all the simple perfect matchings in a complete convex geometric graph on 2m vertices. In particular, we show that all these sets are caterpillar graphs with a special structure, and that their total number is m · 2 m−1 .

Let HD d (p, q) denote the minimal size of a transversal that can always be guaranteed for a family of compact convex sets in R d which satisfy the (p, q)-property (p ≥ q ≥ d + 1). In a celebrated proof of the Hadwiger-Debrunner conjecture, Alon and Kleitman proved that HD d (p, q) exists for all p ≥ q ≥ d + 1. Specifically, they prove that HD d (p, d + 1) isÕ(p d 2 +d ).We present several improved bounds: (i) For any q ≥ d + 1,(iii) For every ǫ > 0 there exists a p 0 = p 0 (ǫ) such that for every p ≥ p 0 and for every q ≥ pThe latter is the first near tight estimate of HD d (p, q) for an extended range of values of (p, q) since the 1957 Hadwiger-Debrunner theorem.We also prove a (p, 2)-theorem for families in R 2 with union complexity below a specific quadratic bound. Based on this, we introduce a polynomial time constant factor approximation algorithm for MAX-CLIQUE of intersection graphs of convex sets satisfying this property.We denote the smallest value s that works for p ≥ q > d by HD d (p, q). The (p, q)-theorem has a rich history of variations and generalizations described in the survey of Eckhoff [Eck03]. Those include a version for set systems with bounded VC-dimension [Mat04], colorful and fractional versions [BFM + 14] and a generalization to a topological (p, q)-theorem for finite families of sets which are so-called good cover, i.e., the intersection of every sub-family is either empty or contractible [AKMM01].The upper bound on HD d (p, q) provided in the Alon-Kleitman proof [AK92] is huge and it is believed that much better bounds could be achieved. In fact, Alon and Kleitman were only interested in proving the existence of HD d (p, q) and hence concentrated on the case q = d + 1. Their bound is HD d (p, d + 1) =Õ(p d 2 +d ) whereÕ hides some polylogarithmic factors. They write: "Although the proof supplies finite upper bounds for HD d (p, q), 1 the bounds obtained are very large and the problem of determining this function precisely remains wide open." In fact, it is not clear how their method can improve the asymptotic of HD d (p, q) when q is slightly more than d + 1, say 2d. In their second paper on the subject [AK97] Alon and Kleitman provide a more elementary proof of the same result that gives slightly weaker bounds.Trivially, for any p ≥ q we have HD d (p, q) ≥ p − q + 1. Hadwiger and Debrunner [HD57] proved the following:The precise bound is not known already in the plane when p = 4 and q = 3. The best known upper and lower bounds in that case are due to Kleitman et al.[KGT01] who showed:

In FOCS '2002, Even et al. introduced and studied the notion of conflict-free colorings of geometrically defined hypergraphs. They motivated it by frequency assignment problems in cellular networks. This notion has been extensively studied since then.A conflict-free coloring of a graph is a coloring of its vertices such that the neighborhood (pointed or closed) of each vertex contains a vertex whose color differs from the colors of all other vertices in that neighborhood. In this paper we study conflictfree colorings of intersection graphs of geometric objects. We show that any intersection graph of n pseudo-discs in the plane admits a conflict-free coloring with O(log n) colors, with respect to both closed and pointed neighborhoods. We also show that the latter bound is asymptotically sharp.Using our methods, we obtain the following strengthening of the two main results of Even et al.: Any family F of n discs in the plane can be colored with O(log n) colors in such a way that for any disc B in the plane, the set of discs in F that intersect B contains a uniquely-colored element. Moreover, such a coloring can be computed deterministically in polynomial time. In view of the original motivation to study such colorings, this strengthening suggests further applications to frequency assignment in wireless networks.Finally, we present bounds on the number of colors needed for conflict-free colorings of other classes of intersection graphs, including intersection graphs of axis-parallel rectangles and of ρ-fat objects in the plane.

In FOCS'2002, Even et al. introduced and studied the notion of conflict-free colorings of geometrically defined hypergraphs. They motivated it by frequency assignment problems in cellular networks. This notion has been extensively studied since then.A conflict-free coloring of a graph is a coloring of its vertices such that the neighborhood (pointed or closed) of each vertex contains a vertex whose color differs from the colors of all other vertices in that neighborhood. In this paper we study conflict-colorings of intersection graphs of geometric objects. We show that any intersection graph of n pseudo-discs in the plane admits a conflict-free coloring with O(log n) colors, with respect to both closed and pointed neighborhoods. We also show that the latter bound is asymptotically sharp. Using our methods, we also obtain a strengthening of the two main results of Even et al. which we believe is of independent interest. In particular, in view of the original motivation to study such colorings, this strengthening suggests further applications to frequency assignment in wireless networks.Finally, we present bounds on the number of colors needed for conflict-free colorings of other classes of intersection graphs, including intersection graphs of axis-parallel rectangles and of ρ-fat objects in the plane.