Hostname: page-component-5b777bbd6c-ks5gx Total loading time: 0 Render date: 2025-06-18T11:09:01.206Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  02 October 2014

ANDRZEJ DUDEK  and
PAWEŁ PRAŁAT
ANDRZEJ DUDEK
Affiliation:
Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008-5248, USA (e-mail: andrzej.dudek@wmich.edu)
PAWEŁ PRAŁAT
Affiliation:
Department of Mathematics, Ryerson University, Toronto, ON, M5B 2K3, Canada (e-mail: pralat@ryerson.ca)
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Primary 05D10Secondary 05C55
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Issue 3 , May 2015 , pp. 551 - 555
Copyright
Copyright © Cambridge University Press 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Alon, N. and Chung, F. R. K. (1988) Explicit construction of linear sized tolerant networks. Discrete Math. 72 15–19.Google Scholar
[2]Beck, J. (1983) On size Ramsey number of paths, trees, and circuits~I. J. Graph Theory 7 115–129.Google Scholar
[3]Beck, J. (1990) On size Ramsey number of paths, trees and circuits~II. In Mathematics of Ramsey Theory (Nešetřil, J. and Rödl, V., eds), Vol. 5 of Algorithms and Combinatorics, Springer, pp. 34–45.Google Scholar
[4]Bollobás, B. (1986) Extremal Graph Theory with Emphasis on Probabilistic Methods, Vol. 62 of CBMS Regional Conference Series in Mathematics, AMS.CrossRefGoogle Scholar
[5]Bollobás, B. (2001) Random Graphs, Cambridge University Press.Google Scholar
[6]Erdős, P. (1981) On the combinatorial problems which I would most like to see solved. Combinatorica 1 25–42.CrossRefGoogle Scholar
[7]Erdős, P., Faudree, R., Rousseau, C. and Schelp, R. (1978) The size Ramsey number. Period. Math. Hungar. 9 145–161.CrossRefGoogle Scholar
[8]Gerencsér, L. and Gyárfás, A. (1967) On Ramsey-type problems. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 10 167–170.Google Scholar
[9]Letzter, S. (2014) Path Ramsey number for random graphs. arXiv:1405.6670 [math.CO]Google Scholar
[10]Pokrovskiy, A. (2014) Partitioning edge-coloured complete graphs into monochromatic cycles and paths. J. Combin. Theory Ser. B 106 70–97.Google Scholar