
Andrzej Ruciński

Eliza Jackowska

Joanna Polcyn
Keywords:
Ramsey numbers, Turan numbers, Hypergraphs
Abstract
Let $P$ denote a 3uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$colored Ramsey number for $P$ is $R(P;r)=r+6$ for $r=2,3$, and that $R(P;r)\le 3r$ for all $r\ge3$. The latter result follows by a standard application of the Turán number $\mathrm{ex}_3(n;P)$, which was determined to be $\binom{n1}2$ in our previous work. We have also shown that the full star is the only extremal 3graph for $P$. In this paper, we perform a subtle analysis of the Turán numbers for $P$ under some additional restrictions. Most importantly, we determine the largest number of edges in an $n$vertex $P$free 3graph which is not a star. These Turántype results, in turn, allow us to confirm the formula $R(P;r)=r+6$ for $r\in\{4,5,6,7\}$.