# 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} }

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.

## 46 Citations

Note on the Multicolour Size-Ramsey Number for Paths,

- MathematicsElectron. J. Comb.
- 2018

This short note gives an alternative proof of the recent result of Krivelevich that $\hat{R}(P_n,r) = O((\log r)r^2 n)$.

On some Multicolor Ramsey Properties of Random Graphs

- MathematicsSIAM J. Discret. Math.
- 2017

It is shown that $5n/2-15/2 \le \hat{R}(P_n) \le 74n$ for $n$ sufficiently large, which improves the previous lower bound and improves the upper bound.

The size-Ramsey number of 3-uniform tight paths

- MathematicsAdvances in Combinatorics
- 2021

Given a hypergraph H, the size-Ramsey number r(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…

The size‐Ramsey number of powers of bounded degree trees

- MathematicsJournal of the London Mathematical Society
- 2020

Given a positive integer s , the s‐colour size‐Ramsey number of a graph H is the smallest integer m such that there exists a graph G with m edges with the property that, in any colouring of E(G) with…

Path Ramsey Number for Random Graphs

- MathematicsCombinatorics, Probability and Computing
- 2015

It is shown that if pn → ∞, w.h.p., whenever G = G(n, p) is 2-edge-coloured there is a monochromatic path of length (2/3 + o(1))n, which is optimal in the sense that 2/3 cannot be replaced by a larger constant.

The size‐Ramsey number of powers of paths

- MathematicsJ. Graph Theory
- 2019

Given graphs G and H and a positive integer q , say that G is q ‐Ramsey for H , denoted G→(H)q , if every q ‐coloring of the edges of G contains a monochromatic copy of H . The size‐Ramsey number…

The size‐Ramsey number of short subdivisions

- MathematicsRandom Struct. Algorithms
- 2021

It is shown that for all constant integers $q,r\geq 2$ and every graph $H$ on $n$ vertices and of bounded maximum degree, the $r$-size-Ramsey number of $H^q$ is at most $(\log n)^{20(q-1)n^{1+1/q}$, for $n $ large enough.

On the size-Ramsey number of grid graphs

- MathematicsComb. Probab. Comput.
- 2021

The size-Ramsey number of a graph F is the smallest number of edges in a graph G with the Ramsey property for F, that is, with the property that any 2-colouring of the edges of G contains a…

Multicolor Size-Ramsey Number of Cycles

- Mathematics
- 2021

Given a positive integer r, the r-color size-Ramsey number of a graph H , denoted by R̂(H, r), is the smallest integer m for which there exists a graph G with m edges such that, in any edge coloring…

On the Size-Ramsey Number of Cycles

- MathematicsCombinatorics, Probability and Computing
- 2019

Various upper bounds for the size-Ramsey numbers of cycles are given, including an alternative proof of ${\hat R_k}({C_n}) \le {c-k}n$ , avoiding use of the regularity lemma.

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