Standard monomials for q-uniform families and a conjecture of Babai and Frankl

Simple groups with the same prime graph as $D_n(q)$

Vasil'ev posed Problem 16.26 in [The Kourovka Notebook: Unsolved Problems in Group Theory, 16th ed.,Sobolev Inst. Math., Novosibirsk (2006).] as follows:Does there exist a positive integer $k$ such that there are no $k$ pairwise nonisomorphicnonabelian finite simple groups with the same graphs of primes? Conjecture: $k = 5$.In [Zvezdina, On nonabelian simple groups having the same prime graph a...

On Set Systems with Restricted Intersections Modulo a Composite Number

Let S be a set of n elements, and let H be a set-system on S, which satisses that the size of any element of H is divisible by m, but the intersection of any two elements of H is not divisible by m. If m is a prime or prime-power, then the famous Frankl{Wilson theorem 3] implies that jHj = O(n m?1), i.e. for xed m, its size is at most polynomial in n. This theorem has numerous applications in c...

Counterexample to the Frankl-Pach Conjecture for Uniform, Dense Families

Covers in Uniform Intersecting Families and a Counterexample to a Conjecture of Lovász

We discuss the maximum size of uniform intersecting families with covering number at least . Among others, we construct a large k-uniform intersecting family with covering number k, which provides a counterexample to a conjecture of Lov asz. The construction for odd k can be visualized on an annulus, while for even k on a Mobius band.

Structure and properties of large intersecting families

We say that a family of k-subsets of an n-element set is intersecting, if any two of its sets intersect. In this paper we study different extremal properties of intersecting families, as well as the structure of large intersecting families. We also give some results on k-uniform families without s pairwise disjoint sets, related to Erdős Matching Conjecture. We prove a conclusive version of Fra...