Tutte's dichromate for signed graphs

We introduce the ``trivariate Tutte polynomial" of a signed graph as an invariant of signed graphs up to vertex switching that contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowhere-zero tensions (for signed graphs they are not simply related to proper colorings as they are for graphs) is given in terms of evaluations of the trivariate Tutte polynomial at two distinct points. Interestingly, the bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share many similar properties with the Tutte polynomial of a graph, does not in general yield the number of nowhere-zero flows of a signed graph. Therefore the ``dichromate" for signed graphs (our trivariate Tutte polynomial) differs from the dichromatic polynomial (the rank-size generating function). The trivariate Tutte polynomial of a signed graph can be extended to an invariant of ordered pairs of matroids on a common ground set -- for a signed graph, the cycle matroid of its underlying graph and its frame matroid form the relevant pair of matroids. This invariant is the canonically defined Tutte polynomial of matroid pairs on a common ground set in the sense of a recent paper of Krajewski, Moffatt and Tanasa, and was first studied by Welsh and Kayibi as a four-variable linking polynomial of a matroid pair on a common ground set.Comment: 53 pp. 9 figure

Big Ramsey degrees of 3-uniform hypergraphs

Given a countably infinite hypergraph $\mathcal R$ and a finite hypergraph $\mathcal A$, the big Ramsey degree of $\mathcal A$ in $\mathcal R$ is the least number $L$ such that, for every finite $k$ and every $k$-colouring of the embeddings of $\mathcal A$ to $\mathcal R$, there exists an embedding $f$ from $\mathcal R$ to $\mathcal R$ such that all the embeddings of $\mathcal A$ to the image $f(\mathcal R)$ have at most $L$ different colours. We describe the big Ramsey degrees of the random countably infinite 3-uniform hypergraph, thereby solving a question of Sauer. We also give a new presentation of the results of Devlin and Sauer on, respectively, big Ramsey degrees of the order of the rationals and the countably infinite random graph. Our techniques generalise (in a natural way) to relational structures and give new examples of Ramsey structures (a concept recently introduced by Zucker with applications to topological dynamics).Comment: 8 pages, 3 figures, extended abstract for Eurocomb 201

Exact big Ramsey degrees via coding trees

We characterize the big Ramsey degrees of free amalgamation classes in finite binary languages defined by finitely many forbidden irreducible substructures, thus refining the recent upper bounds given by Zucker. Using this characterization, we show that the Fra\"iss\'e limit of each such class admits a big Ramsey structure. Consequently, the automorphism group of each such Fra\"iss\'e limit has a metrizable universal completion flow.Comment: Submitted versio

The Removal Property for Linear Configurations in Compact Abelian Groups

The combinatorial removal lemma states that, if a (hyper)graph K has not many copies of the fixed (hyper)graph H , then K can be made free of copies of H by removing a small set of (hyper)edges from K. In this thesis we show results for homomorphisms in finite abelian groups, and for integer linear systems over compact abelian groups that are analogous to the combinatorial removal lemma. The results state that, given some subsets Xi of the group, if there are not many solutions to the system Ax = 0, where the variables xi take values inXi, then there exist small subsets Xi' inside Xi such that there is no solution to the system Ax = 0 with xi in Xi \ Xi'. These results are shown by constructing an appropriate (hyper)graph that allows us to retrieve information on the elements to be removed from Xi using the set of (hyper)edges. These arithmetic removal lemmas extend the first removal lemma for groups proved by Green in 2005. They also present a comprehensive approach to results involving finding non-trivial linear configurations in dense sets. Examples of these types of results are SzemerĂŠdi's Theorem, which ensures the existence of arbitrarily long arithmetic progressions in sets of the integers with positive upper density, or its multidimensional version. The latter was first shown by Furstenberg and Katznelson in 1978 using ergodic theory and proves the existence of k-simplices in subsets containing a positive proportion of the integer lattice or rank k.Ph.D

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

Let G be a finite cyclic group an let r be a positive integer. Let A be a k×m matrix with integer entries. We show that, if A satisfies some natural conditions, then the homogeneous linear system Ax=0 has Ω(|GN|m−k) monochromatic solutions for each r-coloring of GN\{0} and sufficiently large N. Density versions of this counting result are also addressed.Peer ReviewedPostprint (published version

A Tutte Polynomial for Maps

32 pages, 4 figures. Version 2 contains updated grant information for the author A.Goodall (and no other changes)International audienceWe follow the example of Tutte in his construction of the dichromate of a graph (that is, the Tutte polynomial) as a unification of the chromatic polynomial and the flow polynomial in order to construct a new polynomial invariant of maps (graphs embedded in orientable surfaces). We call this the surface Tutte polynomial. The surface Tutte polynomial of a map contains the Las Vergnas polynomial, Bollob\'as-Riordan polynomial and Kruskhal polynomial as specializations. By construction, the surface Tutte polynomial includes among its evaluations the number of local tensions and local flows taking values in any given finite group. Other evaluations include the number of quasi-forests