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Bounding the volume of facet-empty lattice tetrahedra

to be positive if C(i, j) ≥ 0 for every pair (i, j) and C(i, j) is strictly positive for at least one such pair. We prove that the maximum number of ones in an n × n zero-one matrix containing no positive orthogonal cycle is O(n). The order of magnitude of this bound cannot be improved. Our results lead to a new proof of the celebrated theorem of Spencer, Szemerédi, and Trotter [5] stating that the number of times that the unit distance can occur among n points in the plane is O(n). 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 × d rectangle with two semicircles of diameter d attached to its sides of length d. Improving a result of Efrat and Sharir [2], 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.