Turán number of special four cycles in triple systems

Abstract

A special four-cycle F in a triple system consists of four triples inducing a C4. This means that F has four special vertices v1,v2,v3,v4 and four triples in the form wivivi+1 (indices are understood (mod4)) where the wjs are not necessarily distinct but disjoint from {v1,v2,v3,v4}. There are seven non-isomorphic special four-cycles, their family is denoted by F. Our main result implies that the Turán number ex(n,F)=Θ(n3/2). In fact, we prove more, ex(n,{F1,F2,F3})=Θ(n3/2), where the Fi-s are specific members of F. This extends previous bounds for the Turán number of triple systems containing no Berge four cycles.We also study ex(n,A) for all A⊆F. For 16 choices of A we show that ex(n,A)=Θ(n3/2), for 92 choices of A we find that ex(n,A)=Θ(n2) and the other 18 cases remain unsolved.