Abstract. Tuza [ Finite and Infinite Sets, Proc. Colloq. Math. Soc. János Bolyai 37, North Holland, 1981, p. 888] conjectured that [Formula: see text] for all graphs [Formula: see text], where [Formula: see text] is the minimum size of an edge set whose removal makes [Formula: see text] triangle-free and [Formula: see text] is the maximum size of a collection of pairwise edge-disjoint triangles. Here, we generalize Tuza’s conjecture to simple binary matroids that do not contain the Fano plane as a restriction and prove that the geometric version of the conjecture holds for cographic matroids.

A biased graph is a graph $G$, together with a distinguished subset $\mathcal{B}$ of its cycles, so that no theta-subgraph of $G$ contains precisely two cycles in $\mathcal{B}$. A large number of biased graphs can be constructed by choosing $G$ to be a complete graph, and $\mathcal{B}$ to be an arbitrary subset of its Hamilton cycles. We show that, on the logarithmic scale, the total number of simple biased graphs on $n$ vertices does not asymptotically exceed the number that can be constructed in this elementary way.

A matroid is Ingleton if all quadruples of subsets of its ground set satisfy Ingleton's inequality. In particular, representable matroids are Ingleton. We show that the number of Ingleton matroids on ground set $[n]$ is doubly exponential in $n$; it follows that almost all Ingleton matroids are nonrepresentable.

We consider the problem of determining mn, the number of matroids on n elements. The best known lower bound on mn is due to Knuth (1974) who showed that log log mn is at least . On the other hand, Piff (1973) showed that log log mn ≤ n − log n + log log n + O(1), and it has been conjectured since that the right answer is perhaps closer to Knuth's bound. We show that this is indeed the case, and prove an upper bound on log log mn that is within an additive 1 + o(1) term of Knuth's lower bound. Our proof is based on using some structural properties of non-bases in a matroid together with some properties of independent sets in the Johnson graph to give a compressed representation of matroids.