Luis Pedro Montejano

Complete Kneser Transversals

Abstract Let k , d , λ ⩾ 1 be integers with d ⩾ λ . Let m ( k , d , λ ) be the maximum positive integer n such that every set of n points (not necessarily in general position) in R d has the property that the convex hulls of all k -sets have a common transversal ( d − λ ) -plane. It turns out that m ( k , d , λ ) is strongly connected with other interesting problems, for instance, the chromatic number of Kneser hypergraphs and a discrete version of Rados centerpoint theorem. In the same spirit, we introduce a natural discrete version m ⁎ of m by considering the existence of complete Kneser transversals . We study the relation between them and give a number of lower and upper bounds of m ⁎ as well as the exact value in some cases. The main ingredient for the proofs are Radons partition theorem as well as oriented matroids tools. By studying the alternating oriented matroid we obtain the asymptotic behavior of the function m ⁎ for the family of cyclic polytopes.

On Ramsey numbers of complete graphs with dropped stars

Let r ( G , H ) be the smallest integer N such that for any 2 -coloring (say, red and blue) of the edges of K n , n ź N , there is either a red copy of G or a blue copy of H . Let K n - K 1 , s be the complete graph on n vertices from which the edges of K 1 , s are dropped. In this note we present exact values for r ( K m - K 1 , 1 , K n - K 1 , s ) and new upper bounds for r ( K m , K n - K 1 , s ) in numerous cases. We also present some results for the Ramsey number of Wheels versus K n - K 1 , s .

Möbius function of semigroup posets through Hilbert series

In this paper, we investigate the Mobius function µ S associated to a (locally finite) poset arising from a semigroup S of Z m . We introduce and develop a new approach to study µ S by using the Hilbert series of S . The latter enables us to provide formulas for µ S when S belongs to certain families of semigroups. Finally, a characterization for a locally finite poset to be isomorphic to a semigroup poset is given.

Codimension two and three Kneser Transversals

Let

How many circuits determine an oriented matroid

k,d,\lambda \geqslant 1

On the complexity of computing the k-restricted edge-connectivity of a graph

be integers with

Ramsey for complete graphs with a dropped edge or a triangle

d\geqslant \lambda

On the Complexity of Computing the k-restricted Edge-connectivity of a Graph

and let

Connected covering numbers

X

Roudneff's Conjecture for Lawrence Oriented Matroids

be a finite set of points in