Search

A triple system is cancellative if it does not contain three distinct sets A , B , C such that the symmetric difference of A and B is contained in C. We show that every cancellative triple system H that satisfies a particular inequality between the sizes of H and its shadow must be structurally close to the balanced blowup of some Steiner triple system. Our result contains a stability theorem for cancellative triple systems due to Keevash and Mubayi as a special case. It also implies that the boundary of the feasible region of cancellative triple systems has infinitely many local maxima, thus giving the first example showing this phenomenon. [ABSTRACT FROM AUTHOR]

Grosu (2016) [11] asked if there exist an integer r ≥ 3 and a finite family of r -graphs whose Turán density, as a real number, has (algebraic) degree greater than r − 1. In this note we show that, for all integers r ≥ 3 and d , there exists a finite family of r -graphs whose Turán density has degree at least d , thus answering Grosu's question in a strong form. [ABSTRACT FROM AUTHOR]