Richard Wilson conjectured in 1974 the following asymptotic formula for the number of n-vertex Steiner triple systems:The proof is based on the entropy method.As a prelude to this proof we consider the number F(n) of 1-factorizations of the complete graph on n vertices. Using the Kahn-Lovász theorem it can be shown that F(n) ≤ (1 + o(1)) n e 2 n 2 2 .We show how to derive this bound using the entropy method. Both bounds are conjectured to be sharp.

What is the higher-dimensional analog of a permutation? If we think of a permutation as given by a permutation matrix, then the following definition suggests itself: A d-dimensional permutation of order n is an n × n × . . . n = [n] d+1 array of zeros and ones in which every line contains a unique 1 entry. A line here is a set of entries of the form {(x 1 , . . . , x i−1 , y, x i+1 , . . . , x d+1 )|n ≥ y ≥ 1} for some index d + 1 ≥ i ≥ 1 and some choice of x j ∈ [n] for all j = i. It is easy to observe that a one-dimensional permutation is simply a permutation matrix and that a two-dimensional permutation is synonymous with an order-n Latin square. We seek an estimate for the number of ddimensional permutations. Our main result is the following upper bound on their number.

In how many ways can n queens be placed on an n × n chessboard so that no two queens attack each other? This is the famous n-queens problem. Let Q(n) denote the number of such configurations, and let T (n) be the number of configurations on a toroidal chessboard. We show that for every n of the form 4 k + 1, T (n) and Q(n) are both at least n Ω(n) . This result confirms a conjecture of Rivin, Vardi and Zimmerman for these values of n [11]. We also present new upper bounds on T (n) and Q(n) using the entropy method, and conjecture that in the case of T (n) the bound is asymptotically tight. Along the way, we prove an upper bound on the number of perfect matchings in regular hypergraphs, which may be of independent interest.

Abstract. Let T (n) denote the maximal number of transversals in an order-n Latin square. Improving on the bounds obtained by McKay et al., Taranenko recently proved that T (n) ≤ (1 + o(1)) n e 2 n , and conjectured that this bound is tight.We prove via a probabilistic construction that indeed T (n) = (1 + o(1)) n e 2 n .Until the present paper, no superexponential lower bound for T (n) was known. We also give a simpler proof of the upper bound.

Abstract. Let L be an order-n Latin square. For X, Y, Z ⊆ {1, ..., n}, let L(X, Y, Z) be the number of triples i ∈ X, j ∈ Y, k ∈ Z such that L(i, j) = k. We conjecture that asymptotically almost every Latin square satisfiesholds asymptotically almost surely (this bound is obviously tight). We show that there exist Latin squares with ε(L) = O(n 2 ), and that ε(L) = O(n 2 log 2 n) for almost every order-n Latin square. On the other hand, we recall that ε(L) ≥ Ω(n 33/14 ) if L is the multiplication table of an order-n group. We also show the existence of Latin squares in which every empty cube has side length O((n log n) 1/2 ), which is tight up to the √ log n factor. Some of these results extend to higher dimensions. Many open problems remain.