Multicolor Size-Ramsey Number of Paths

Abstract

The size-Ramsey number of a graph denoted by is the smallest integer such that there is a graph with edges with this property that for any coloring of the edges of with colors, contains a monochromatic copy of. The investigation of the size-Ramsey numbers of graphs was initiated by Erdős‚ Faudree‚ Rousseau and Schelp in 1978. Since then, Size-Ramsey numbers have been studied with particular focus on the case of trees and bounded degree graphs. Addressing a question posed by Erdős‚ Beck [2] proved that the size-Ramsey number of the path is linear in by means of a probabilistic construction. In fact, Beck’s proof implies that and this upper bound was improved several times. Currently‚ the best known upper bound is due to Dudek and Prałat [4] which proved that . On the other hand‚ the first nontrivial lower bound for was provided by Beck and his result was subsequently improved by Dudek and Prałat [3] who showed that. The strongest known lower bound was proved recently by Bal and DeBiasio [1]. ./files/site1/files/%D8%AC%D9%88%D8%A7%D8%AF%DB%8C_%D9%85%DB%8C%D8%B1%D8%B9%D9%84%D8%A7%DB%8C%DB%8C.pdf

Keywords

• ramsey number
• size ramsey number
• path