Geodesic
The size-Ramsey number of 3-uniform tight paths
Advances in Combinatronics (2021)

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Given a hypergraph $H$, the size-Ramsey number $\hat{r}_2(H)$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges with the property that in any colouring of the edges of $G$ with two colours there is a monochromatic copy of $H$. We prove that the size-Ramsey number of the $3$-uniform tight path on $n$ vertices $P^{(3)}_n$ is linear in $n$, i.e., $\hat{r}_2(P^{(3)}_n) = O(n)$. This answers a question by Dudek, Fleur, Mubayi, and R\"odl for $3$-uniform hypergraphs [On the size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016), 417-434], who proved $\hat{r}_2(P^{(3)}_n) = O(n^{3/2} \log^{3/2} n)$.
Publié le : 
@article{ADVC_2021_a4,
     author = {Jie Han and Yoshiharu Kohayakawa and Shoham Letzter and Guilherme Oliveira Mota and Olaf Parczyk},
     title = {The {size-Ramsey} number of 3-uniform tight paths},
     journal = {Advances in Combinatronics},
     publisher = {mathdoc},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADVC_2021_a4/}
}
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%J Advances in Combinatronics
%D 2021
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Jie Han; Yoshiharu Kohayakawa; Shoham Letzter; Guilherme Oliveira Mota; Olaf Parczyk. The size-Ramsey number of 3-uniform tight paths. Advances in Combinatronics (2021). http://geodesic.mathdoc.fr/item/ADVC_2021_a4/