A typical problem in extremal combinatorics is the following. Given a large number n and a set L, find the maximum cardinality of a family of subsets of a ground set of n elements such that the intersection of any two subsets has cardinality in L. We investigate the generalization of this problem, where intersections of more than 2 subsets are considered. In particular, we prove that when k − 1...

Let F/Q be a totally real number field of degree n. We explicitly evaluate a certain sum of rational functions over a infinite fan of F -rational polyhedral cones in terms of the norm map N:F → Q. This completes Sczech’s combinatorial proof of Satake’s conjecture connecting the special values of L-series associated to cusp singularities with intersection numbers of divisors in their toroidal re...

Several classical constructions illustrate the fact that the chromatic number of a graph may be arbitrarily large compared to its clique number. However, until very recently no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set X in R that is not an axis-aligned rectangle and for any pos...

1.1. This is the first of two papers addressing a Cauchy type inequality for the geometric intersection number between two 1-dimensional submanifolds in a surface. As a consequence, we reestablish some of the basic results in Thurston's theory of measured laminations. In this paper, we consider surfaces with non-empty boundary using ideal triangulations. In the sequel, we establish the inequali...

In this paper we reduce the problem concerning the number of connected components in the intersection of two real opposite open Schubert cells in SLn(R)/B to a purely combinatorial question in the space of upper triangular matrices with F2-valued entries. The crucial step of the reduction uses the parametrization of the space of real unipotent totally positive upper triangular matrices introduc...

We propose an algorithm based on Newton’s method and subdivision for finding all zeros of a polynomial system in a bounded region of the plane. This algorithm can be used to find the intersections between a line and a surface, which has applications in graphics and computer-aided geometric design. The algorithm can operate on polynomials represented in any basis that satisfies a few conditions....

Quasi-symmetric 3-designs with block intersection numbers x, y (0 ≤ x ≤ y < k) are studied. It is proved that the parameter λ of a quasi-symmetric 3-(v, k, λ) design satisfies a quadratic whose coefficients are polynomial functions of k, x and y. We use this quadratic to prove that there exist finitely many quasi-symmetric 3-designs under either of the following two restrictions: 1. The block i...

Suppose that is an innnite set and k is a natural number. Let ] k denote the set of all k-subsets of and let F be a eld. In this paper we study the FSym(()-submodule structure of the permutation module F] k. Using the representation theory of nite symmetric groups, we show that every submodule of F] k can be written as an intersection of kernels of certain FSym(()-homomorphisms F] k ?! F] l for...

The matroid parity problem is a generalization of rnatroid intersection and general graph matching (and hence network flow, degree-constrained subgraphs, etc.). A polynomial algorithm for linear matroids was presented by L O W l j Z . This paper presents an algorithm that uses time O(nzn3). where m is the number of elements and n is tk,e rank; for the spanning tree parity problem the time ,a O(...

We show that every complete intersection of Laurent polynomials in an algebraic torus is isomorphic to a complete intersection of master functions in the complement of a hyperplane arrangement, and vice versa. We call this association Gale duality because the exponents of the monomials in the polynomials annihilate the weights of the master functions and linear forms defining the two systems al...