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 n− 3 2 log n−O(1). 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 stable sets in the Johnson graph to give a compressed representation of matroids.

To evaluate the risk of abortion after genetic amniocentesis in twin pregnancies, a retrospective study of 15 centers was performed. The spontaneous abortion rate up to 20 completed weeks of gestation was 2.3%; the abortion rate up to 28 completed weeks, as defined by WHO, was 3.7%. The abortion rate could not be correlated either with the number of needle insertions or with the type of marker dye used. There was also no correlation between the abortion rate and the gestational age at which amniocentesis was performed. A significant association was shown between congenital intestinal obstructions and the application of methylene blue intra-amniotically as a marker dye. Considering the increased natural loss rate in multiple gestations, amniocentesis in twin pregnancies seems to be a safe and reliable technique.

A flat cover is a collection of flats identifying the non-bases of a matroid. We introduce the notion of cover complexity, the minimal size of such a flat cover, as a measure for the complexity of a matroid, and present bounds on the number of matroids on n elements whose cover complexity is bounded. We apply cover complexity to show that the class of matroids without an N -minor is asymptotically small in case N is one of the sparse paving matroids U 2,k , U 3,6 , P 6 , Q 6 or R 6 , thus confirming a few special cases of a conjecture due to Mayhew, Newman, Welsh, and Whittle. On the other hand, we show a lower bound on the number of matroids without an M (K 4 )-minor which asymptotically matches the best known lower bound on the number of all matroids, due to Knuth.

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 non-representable.

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