A Bound of the Cardinality of Families Not Containing \(\Delta \) -Systems

P. Erdős and R. Rado defined a \(\Delta \)-system as a family in which every two members have the same intersection. Here we obtain a new upper bound of the maximum cardinality \(\varphi (n)\) of an n-uniform family not containing any \(\Delta \)-system of cardinality 3. Namely, we prove that for any α > 1, there exists C = C(α) such that for any n, $$\displaystyle{\varphi (n) \leq Cn{!\alpha }^{-n}.}$$