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}
}
  • A. Dudek, P. Prałat
  • 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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