# Weak \(\Delta\)-systems

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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

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