On Systems of Small Sets with No Large Δ-Subsystems

Source abstract

A family of k sets is called a Δ- system if any two sets have the same intersection. Denote by f ( r , k ) the least integer so that any r -uniform family of f ( r , k ) sets contains a Δ-system consisting of k sets. We prove that, for every fixed r , f ( r , k ) = k r + o ( k r ). Using a recent result of Molloy and Reed [5], a bound on the error term is provided for sufficiently large k .