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- W2224556402 abstract "Let $P$ denote a 3-uniform 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 $rge3$. The latter result follows by a standard application of the Tur'an number $ex_3(n;P)$, which was determined to be $binom{n-1}2$ in our previous work. We have also shown that the full star is the only extremal 3-graph for $P$. In this paper, we perform a subtle analysis of the Tur'an numbers for $P$ under some additional restrictions. Most importantly, we determine the largest number of edges in an $n$-vertex $P$-free 3-graph which is not a star. These Tur'an type results, in turn, allow us to confirm the formula $R(P;r)=r+6$ for $rin{4,5,6,7}$." @default.
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- W2224556402 date "2015-07-15" @default.
- W2224556402 modified "2023-09-27" @default.
- W2224556402 title "Multicolor Ramsey numbers and restricted Tur'an numbers for the loose 3-uniform path of length three" @default.
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