A discrete isodiametric result: the Erd\H{o}s-Ko-Rado theorem for multisets

There are many generalizations of the Erd\H{o}s-Ko-Rado theorem. We give new results (and problems) concerning families of $t$-intersecting $k$-element multisets of an $n$-set and point out connections to coding theory and classical geometry. We establish the conjecture that for $n \geq t(k-t)+2$ such a family can have at most ${n+k-t-1\choose k-t}$ members

Matroid toric ideals: complete intersection, minors and minimal systems of generators

In this paper, we investigate three problems concerning the toric ideal associated to a matroid. Firstly, we list all matroids $\mathcal M$ such that its corresponding toric ideal $I_{\mathcal M}$ is a complete intersection. Secondly, we handle with the problem of detecting minors of a matroid $\mathcal M$ from a minimal set of binomial generators of $I_{\mathcal M}$. In particular, given a minimal set of binomial generators of $I_{\mathcal M}$ we provide a necessary condition for $\mathcal M$ to have a minor isomorphic to $\mathcal U_{d,2d}$ for $d \geq 2$. This condition is proved to be sufficient for $d = 2$ (leading to a criterion for determining whether $\mathcal M$ is binary) and for $d = 3$. Finally, we characterize all matroids $\mathcal M$ such that $I_{\mathcal M}$ has a unique minimal set of binomial generators.Comment: 9 page

On generalized Kneser hypergraph colorings

In Ziegler (2002), the second author presented a lower bound for the chromatic numbers of hypergraphs \KG{r}{\pmb s}{\calS}, "generalized $r$-uniform Kneser hypergraphs with intersection multiplicities $\pmb s$." It generalized previous lower bounds by Kriz (1992/2000) for the case ${\pmb s}=(1,...,1)$ without intersection multiplicities, and by Sarkaria (1990) for \calS=\tbinom{[n]}k. Here we discuss subtleties and difficulties that arise for intersection multiplicities $s_i>1$: 1. In the presence of intersection multiplicities, there are two different versions of a "Kneser hypergraph," depending on whether one admits hypergraph edges that are multisets rather than sets. We show that the chromatic numbers are substantially different for the two concepts of hypergraphs. The lower bounds of Sarkaria (1990) and Ziegler (2002) apply only to the multiset version. 2. The reductions to the case of prime $r$ in the proofs Sarkaria and by Ziegler work only if the intersection multiplicities are strictly smaller than the largest prime factor of $r$. Currently we have no valid proof for the lower bound result in the other cases. We also show that all uniform hypergraphs without multiset edges can be represented as generalized Kneser hypergraphs.Comment: 9 pages; added examples in Section 2; added reference ([11]), corrected minor typos; to appear in J. Combinatorial Theory, Series

Anagram-free Graph Colouring

An anagram is a word of the form $WP$ where $W$ is a non-empty word and $P$ is a permutation of $W$. We study anagram-free graph colouring and give bounds on the chromatic number. Alon et al. (2002) asked whether anagram-free chromatic number is bounded by a function of the maximum degree. We answer this question in the negative by constructing graphs with maximum degree 3 and unbounded anagram-free chromatic number. We also prove upper and lower bounds on the anagram-free chromatic number of trees in terms of their radius and pathwidth. Finally, we explore extensions to edge colouring and $k$-anagram-free colouring.Comment: Version 2: Changed 'abelian square' to 'anagram' for consistency with 'Anagram-free colourings of graphs' by Kam\v{c}ev, {\L}uczak, and Sudakov. Minor changes based on referee feedbac

A Comparison of Well-Quasi Orders on Trees

Well-quasi orders such as homeomorphic embedding are commonly used to ensure termination of program analysis and program transformation, in particular supercompilation. We compare eight well-quasi orders on how discriminative they are and their computational complexity. The studied well-quasi orders comprise two very simple examples, two examples from literature on supercompilation and four new proposed by the author. We also discuss combining several well-quasi orders to get well-quasi orders of higher discriminative power. This adds 19 more well-quasi orders to the list.Comment: In Proceedings Festschrift for Dave Schmidt, arXiv:1309.455