Turán numbers $T(n,5,3)$ and graphs without induced $5$-cycles

Abstract

Turán number $T(n,5,3)$ is the minimum size of a system of triples out of a base set $X$ of $n$ elements such that every quintuple in $X$ contains a triple from the system. The exact values of $T(n,5,3)$ are known for $n\le 17$. Turán conjectured that $T(2m,5,3)=2\left(\genfrac{}{}{0ex}{}{m}{3}\right)$, and no counterexamples have been found so far. If this conjecture is true, then $T(2m+1,5,3)\ge \lceil m(m-2)(2m+1)/6\rceil$. We prove the matching upper bound for all $n=2m+1>17$ except $n=27$.