Let G be a semi-simple Lie group with finite center and S ⊂ G a semigroup with intS 6= Ø . A closed subgroup L ⊂ G is said to be S -admissible if S is transitive in G/L . In [10] it was proved that a necessary condition for L to be S -admissible is that its action in B (S) is minimal and contractive where B (S) is the flag manifold associated with S , as in [9]. It is proved here, under an addi...

In 1973 Paul Erdős conjectured that there is an integer v0(r) such that, for every v > v0(r) and v ≡ 1,3 (mod 6), there exists a Steiner triple system of order v, containing no i blocks on i + 2 points for every 1 < i ≤ r . Such an STS is said to be r-sparse. In this paper we consider relations of automorphisms of an STS to its sparseness. We show that for every r ≥ 13 there exists no point-tra...

If T is an n-vertex tournament with a given number of 3-cycles, what can be said about the number of its 4-cycles? The most interesting range of this problem is where T is assumed to have c ⋅ n cyclic triples for some c > 0 and we seek to minimize the number of 4-cycles. We conjecture that the (asymptotic) minimizing T is a random blow-up of a constant-sized transitive tournament. Using the met...