# An Alternative Proof of the Linearity of the Size-Ramsey Number of Paths

@article{Dudek2014AnAP,
title={An Alternative Proof of the Linearity of the Size-Ramsey Number of Paths},
author={Andrzej Dudek and Paweł Prałat},
journal={Combinatorics, Probability and Computing},
year={2014},
volume={24},
pages={551 - 555}
}
• Published 7 May 2014
• Mathematics
• Combinatorics, Probability and Computing
The size-Ramsey number $\^{r}$(F) of a graph F is the smallest integer m such that there exists a graph G on m edges with the property that every colouring of the edges of G with two colours yields a monochromatic copy of F. In 1983, Beck provided a beautiful argument that shows that $\^{r}$(Pn) is linear, solving a problem of Erdős. In this note, we provide another proof of this fact that actually gives a better bound, namely, $\^{r}$(Pn) < 137n for n sufficiently large.
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Subdivisions Contractions Small graphs of large girth Large graphs of small diameter Cycles in dense graphs The evolution of random graphs The size Ramsey number of a path Weakly saturated graphs