Constructions of Tur\'an systems that are tight up to a multiplicative constant

Abstract

For positive integers $n\ge s> r$, the Tur\'an 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 Tur\'an density $t(s,r)$ as the limit of $T(n,s,r)/ {n\choose r}$ as $n\to\infty$. The question of estimating these parameters received a lot of attention after it was first raised by Tur\'an in 1941. A trivial lower bound is $t(s,r)\ge 1/{s\choose s-r}$. In the early 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 $R\ge1$ there is $\mu_R$ (in fact, $\mu_R$ can be taken to grow as $(1+o(1))\, R\ln R$) such that $t(r+R,r)\le (\mu_R+o(1))/ {r+R\choose R}$ as $r\to\infty$, that is, the trivial lower bound is tight for every $R$ up to a multiplicative constant $\mu_R$.