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

The Turán number T(n,5,3) $T(n,5,3)$ is the minimum size of a system of triples out of a base set X $X$ of n $n$ elements such that every quintuple in X $X$ contains a triple from the system. The exact values of T(n,5,3) $T(n,5,3)$ are known for n≤17 $n\le 17$. Turán conjectured that T(2m,5,3)=2m3 $T(2m,5,3)=2\left(\genfrac{}{}{0.0pt}{}{m}{3}\right)$, and no counterexamples have been found so far. If this conjecture is true, then T(2m+1,5,3)≥⌈m(m−2)(2m+1)∕6⌉ $T(2m+1,5,3)\ge \lceil m(m-2)(2m+1)\unicode{x02215}6\rceil $. We prove the matching upper bound for all n=2m+1>17 $n=2m+1\gt 17$ except n=27 $n=27$. [ABSTRACT FROM AUTHOR]