Forbidden patterns and unit distances

• Authors: János Pach;Gábor Tardos
• Venue: SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
• Year: 2005

At most how many edges (hyperedges, nonzero entries, characters) can a graph (hypergraph, zero-one matrix, string) have if it does not contain a fixed forbidden pattern? Turán-type extremal graph theory, Erdős--Ko--Rado-type extremal set theory, Ramsey theory, the theory of Davenport--Schinzel sequences, etc. have been developed to address questions of this kind. They produced a number of results that found important applications in discrete and computational geometry.In the present paper, we discuss an extension of extremal graph theory to ordered graphs, i.e., to graphs whose vertex set is linearly ordered. In the most interesting cases, the forbidden ordered graphs are bipartite, and the basic problem can be reformulated as an extremal problem for zero-one matrices avoiding a certain submatrix P. We disprove a general conjecture of Füredi and Hajnal related to the latter problem, and replace it by some weaker alternatives. We verify our conjectures in a few special cases when P is the adjacency matrix of an acyclic graph and discuss the same question when the forbidden patterns are adjacency matrices of cycles.Our results lead to a new proof of the celebrated theorem of Spencer, Szemerédi, and Trotter [15] stating that the number of times that the unit distance can occur among n points in the plane is O(n4/3). This is the first proof that does not use any tool other than a forbidden pattern argument. We present another geometric application, where the forbidden pattern P is the adjacency matrix of an acyclic graph. A hippodrome is a c x d rectangle with two semidisks of diameter d attached to its sides of length d. Improving a result of Efrat and Sharir [5] we show that the number of "free" placements of a convex n-gon in general position in a hippodrome H such that simultaneously three vertices of the polygon lie on the boundary of H, is O(n). This result is related to the Planar Segment-Center Problem.