Some heterochromatic theorems for matroids

The anti-Ramsey number of Erdös, Simonovits and Sós from 1973 has become a classic invariant in Graph Theory. To extend this invariant to Matroid Theory, we use the heterochromatic number $hc\left(H\right)$ of a non-empty hypergraph $H$. The heterochromatic number of $H$ is the smallest integer $k$ such that for every colouring of the vertices of $H$ with exactly $k$ colours, there is a totally multicoloured hyperedge of $H$.Given a matroid $M$, there are several hypergraphs over the ground set of $M$ we can consider, for example, $C\left(M\right)$, whose hyperedges are the circuits of $M$, or $B\left(M\right)$, whose hyperedges are the bases of $M$. We determine $hc\left(C\left(M\right)\right)$ for general matroids and characterise the matroids with the property that $hc\left(B\left(M\right)\right)$ equals the rank of the matroid. We also consider the case when the hyperedges are the Hamiltonian circuits of the matroid. Finally, we extend the known result about the anti-Ramsey number of 3-cycles in complete graphs to the heterochromatic number of 3-circuits in projective geometries over finite fields, and we propose a problem very similar to the famous problem on the anti-Ramsey number of the $p$-cycles.     

Infinite families of 2-designs and 3-designs from linear codes

The interplay between coding theory and $t$-designs started many years ago. While every $t$-design yields a linear code over every finite field, the largest $t$ for which an infinite family of $t$-designs is derived directly from a linear or nonlinear code is $t=3$. Sporadic $4$-designs and $5$-designs were derived from some linear codes of certain parameters. The major objective of this paper is to construct many infinite families of $2$-designs and $3$-designs from linear codes. The parameters of some known $t$-designs are also derived. In addition, many conjectured infinite families of $2$-designs are also presented.     

Clones in matroids representable over a prime field

We show that for every prime number $p$, a 3-connected non-uniform $GF\left(p\right)$-representable matroid can have a clone set of size at most $p?2$.     

Unipancyclic matroids

Analogous to the concept of uniquely pancyclic graphs, we define a uniquely pancyclic (UPC) matroid of rank $r$ to be a (simple) rank-$r$ matroid containing exactly one circuit of each length $?$ for $3\le ?\le r+1$. Our discussion addresses the existence of graphic, binary, and transversal representations of UPC matroids. Using Shi’s results, which catalogued exactly seven non-isomorphic UPC graphs, we produce a nongraphic binary UPC matroid of rank 24. We consider properties of binary UPC matroids in general, and prove that all binary UPC matroids have a connectivity of $2$.     

Deletion–contraction to form a polymatroid

Let $M$ be a matroid with rank function $r$, and let $e\in E\left(M\right)$. The deletion–contraction polymatroid with rank function $f=r_{M?e}+r_{M/e}$ will be denoted $P_e\left(M\right)$. Notice that $P_e\left(M\right)$ is uniquely determined by $M$ and $e$. Similarly, a deletion–contraction polymatroid determines $M$, unless $e$ is a loop or co-loop. This paper will characterize all polymatroids of this deletion–contraction form by giving the set of excluded minors. Vertigan conjectured that the class of $GF\left(q\right)$-representable deletion–contraction polymatroids is well-quasi-ordered. From this attractive conjecture, both Rota’s Conjecture and the WQO Conjecture for $GF\left(q\right)$-representable matroids would follow.     

The number of lines in a matroid with no U2,n-minor

We show that, if $q$ is a prime power at most $5$, then every rank-$r$ matroid with no $U_{2,q+2}$-minor has no more lines than a rank-$r$ projective geometry over $\mathrm{GF}\left(q\right)$. We also give examples showing that for every other prime power this bound does not hold.