Constructions of Turán systems that are tight up to a multiplicative constant
Oleg Pikhurko
TL;DR
The paper investigates Turán $(n,s,r)$-systems and their asymptotic density $t(s,r)$, establishing that the trivial lower bound is tight up to a multiplicative constant for all fixed $R$ by constructing $t(r+R,r)\le (\mu_R+o(1))/{{r+R}\choose R}$ with $\mu_R=(1+o(1))R\ln R$. It introduces a recursive probabilistic framework that builds Turán $(n,r+R,r)$-systems via extensions $S^*$ and $T^*$ and a base construction $H_n^r$, yielding explicit constants for the base case $t(r+1,r)$ (e.g., $t(n,r+1,r)\le 6.239/(r+1)$ and $\le 4.911/(r+1)$ for large $r$). The analysis hinges on choosing parameters $\beta,c,\mu$ and a root $c_0$ of $e^{c}=(c+1)^{R+1}$ to optimize the bound, and extends to general $R$ with a small additive term $D/\ln(r+3)$. These results resolve a long-standing conjecture (disproving de Caen's $rt(r+1,r)\to\infty$) and advance extremal hypergraph theory by demonstrating near-tightness relative to the trivial lower bound and providing constructive approaches with potential algorithmic applications.
Abstract
For positive integers $n\ge s> r$, 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 $T(n,s,r)/ {n\choose r}$ 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 $t(s,r)\ge 1/{s\choose s-r}$. 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 $μ_R$ (in fact, $μ_R$ can be taken to grow as $(1+o(1))\, R\ln R$) such that $t(r+R,r)\le (μ_R+o(1))/ {r+R\choose R}$ as $r\to\infty$, that is, the trivial lower bound is tight for every $R$ up to a multiplicative constant $μ_R$.