Multicolor Size-Ramsey Number of Paths

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 .