# Weak $$\Delta$$-systems

A hypergraph $$H$$ consists of a vertex set $$V$$ together with a family $$E$$ of subsets of $$V$$, which are called the edges of $$H$$. A $$r$$-uniform hypergraph, or $$r$$-graph, for short, is a hypergraph whose edge set consists of $$r$$-subsets of $$V$$. A graph is just a special case of an $$r$$-graph with $$r=2$$.

A family $${\mathcal A} = (A_1, \ldots, A_s)$$ is called a weak $$\Delta$$-system if $$A_i \cap A_j$$, for $$i \not = j$$, are all of the same size. Let $$g(n,k)$$ denote the least size for a family of n-sets forcing a weak $$\Delta$$-system of $$k$$ sets. Erdös, Milner and Rado [2] proposed the following problem:

# A conjecture on weak $$\Delta$$-systems $g (n,3) < c^n.$

Recently, Axenovich, Fon der Flaass and Kostochka [1] proved

$$\displaystyle g(n,3) < (n!)^{1/2+ \epsilon}.$$