Beyond chromatic threshold via (p,q)-theorem, and blow-up phenomenon

We establish a novel connection between the well-known chromatic threshold problem in extremal combinatorics and the celebrated (p,q)-theorem in discrete geometry. In particular, for a graph G with bounded clique number and a natural density condition, we prove a (p,q)-theorem for an abstract convexity space associated with G. Our result strengthens those of Thomassen and Nikiforov on the chromatic threshold of cliques. Our (p,q)-theorem can also be viewed as a χ-boundedness result for (what we call) ultra maximal Kr-free graphs. We further show that the graphs under study are blow-ups of constant size graphs, improving a result of Oberkampf and Schacht on homomorphism threshold of cliques. Our result unravels the cause underpinning such a blow-up phenomenon, differentiating the chromatic and homomorphism threshold problems for cliques. Our result implies that for the homomorphism threshold problem, rather than the minimum degree condition usually considered in the literature, the decisive factor is a clique density condition on co-neighborhoods of vertices. More precisely, we show that if an n-vertex Kr-free graph G satisfies that the common neighborhood of every pair of non-adjacent vertices induces a subgraph with Kr-2-density at least ε > 0, then G must be a blow-up of some Kr-free graph F on at most 2O (r/ε log 1/ε) vertices. Furthermore, this single exponential bound is optimal. We construct examples with no Kr-free homomorphic image of size smaller than 2Ωr (1/ε)

Combinatorial properties of non-archimedean convex sets

We study combinatorial properties of convex sets over arbitrary valued fields. We demonstrate analogs of some classical results for convex sets over the reals (e.g. the fractional Helly theorem and B\'ar\'any's theorem on points in many simplices), along with some additional properties not satisfied by convex sets over the reals, including finite breadth and VC-dimension. These results are deduced from a simple combinatorial description of modules over the valuation ring in a spherically complete valued field.Comment: v.2: 27 pages; some minor corrections following referees' reports; added a brief discussion of the other notions of convexity in valued fields (Section 5.2) and connections to the study of abstract convexity spaces (Section 5.3); accepted to the Pacific Journal of Mathematic

Approximating Maximum Diameter-Bounded Subgraph in Unit Disk Graphs

We consider a well studied generalization of the maximum clique problem which is defined as follows. Given a graph G on n vertices and an integer d >= 1, in the maximum diameter-bounded subgraph problem (MaxDBS for short), the goal is to find a (vertex) maximum subgraph of G of diameter at most d. For d=1, this problem is equivalent to the maximum clique problem and thus it is NP-hard to approximate it within a factor n^{1-epsilon}, for any epsilon > 0. Moreover, it is known that, for any d >= 2, it is NP-hard to approximate MaxDBS within a factor n^{1/2 - epsilon}, for any epsilon > 0. In this paper we focus on MaxDBS for the class of unit disk graphs. We provide a polynomial-time constant-factor approximation algorithm for the problem. The approximation ratio of our algorithm does not depend on the diameter d. Even though the algorithm itself is simple, its analysis is rather involved. We combine tools from the theory of hypergraphs with bounded VC-dimension, k-quasi planar graphs, fractional Helly theorems and several geometric properties of unit disk graphs