Tur\'{a}n numbers $T(n,5,3)$ and graphs without induced $5$-cycles

TL;DR

This paper investigates the Turán number T(n,5,3), providing new bounds and proofs for its exact values for larger n, and explores the structure of graphs avoiding induced 5-cycles.

Contribution

The paper proves a matching upper bound for T(n,5,3) for all odd n > 17 except 27, advancing understanding of Turán numbers and graphs without induced 5-cycles.

Findings

Established exact values of T(n,5,3) for n > 17 except 27.

Proved the Turán conjecture for most larger n.

Provided bounds for T(2m+1,5,3) based on conjectured formulas.

Abstract

Tur\'{a}n 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 $n \leq 17$. Tur\'{a}n conjectured that $T(2m,5,3) = 2\binom{m}{3}$, and no counterexamples have been found so far. If this conjecture is true, then $T(2m+1,5,3) \geq \lceil m(m-2)(2m+1)/6\rceil$. We prove the matching upper bound for all $n = 2m+1 > 17$ except $n=27$.