Tur\'an and Ramsey numbers for $3$-uniform minimal paths of length $4$

TL;DR

This paper determines Turán numbers for 3-uniform minimal paths of length four across all sizes and uses these to compute related Ramsey numbers for up to four colors, advancing extremal combinatorics knowledge.

Contribution

It provides exact Turán numbers for 3-uniform minimal paths of length four for all n and calculates associated Ramsey numbers for multiple colors, filling a gap in combinatorial extremal theory.

Findings

Exact Turán numbers for all n

Second and third order Turán numbers established

Ramsey numbers for up to four colors computed

Abstract

We determine Tur\'an numbers for the family of 3-uniform minimal paths of length four \emph{for all $n$}. We also establish the second and third order Tur\'an numbers and use them to compute the corresponding Ramsey numbers for up to four colors.