Constructions of Turán systems that are tight up to a multiplicative constant

Abstract

For positive integers $n⩾s>r$, the Turán function $T(n,s,r)$ is the smallest size of an r-graph with n vertices such that every set of s vertices contains at least one edge. Also, define the Turán density $t(s,r)$ as the limit of $T(n,s,r)/\left(\begin{array}{c}n\\ r\end{array}\right)$ as $n\to \infty$. The question of estimating these parameters received a lot of attention after it was first raised by Turán in 1941. A trivial lower bound is $t(s,r)⩾1/\left(\begin{array}{c}s\\ s-r\end{array}\right)$. In the 1990s, de Caen conjectured that $r\cdot t(r+1,r)\to \infty$ as $r\to \infty$ and offered 500 Canadian dollars for resolving this question.

We disprove this conjecture by showing more strongly that for every integer $R⩾1$ there is ${\mu }_R$ (in fact, ${\mu }_R$ can be taken to grow as $(1+o(1\left)\right)R\ln R$) such that $t(r+R,r)⩽({\mu }_R+o(1\left)\right)/\left(\begin{array}{c}r+R\\ R\end{array}\right)$ as $r\to \infty$, that is, the trivial lower bound is tight for every R up to a multiplicative constant ${\mu }_R$.