Lluís Vena

On the number of monochromatic solutions of integer linear systems on abelian groups

Let G be a finite abelian group with exponent n, and let r be a positive integer. Let A be a kxm matrix with integer entries. We show that if A satisfies some natural conditions and |G| is large enough then, for each r-coloring of G@?{0}, there is @d depending only on r,n and m such that the homogeneous linear system Ax=0 has at least @d|G|^m^-^k monochromatic solutions. Density versions of this counting result are also addressed.

A Tutte polynomial for non-orientable maps

We construct a new polynomial invariant of maps (graphs embedded in closed surfaces, not necessarily orientable). Our invariant is tailored to contain as evaluations the number of local flows and local tensions taking non-identity values in any given finite group. Moreover, it contains as specializations the Krushkal polynomial, the Bollobas-Riordan polynomial, the Las Vergnas polynomial, and their extensions to non-orientable surfaces, and hence in particular the Tutte polynomial of the under-lying graph of the map.

On the chromatic number of a subgraph of the Kneser graph

Let

Counting configuration-free sets in groups

n

Matroid invariants and counting graph homomorphisms

and

On limits of sparse random graphs

k

Counting configuration–free sets in groups

be positive integers with

On the number of monochromatic solutions of integer linear systems on Abelian groups

n \geq 2k

On the removal lemma for linear systems over Abelian groups

. Consider a circle

A Tutte Polynomial for Maps

C