عدد رمزي اندازه اي چند رنگي مسيرها

Multicolor Size-Ramsey Number of Paths

عدد رمزي اندازه اي 7 رنگي گراف F كه با نماد (F,r) نشان داده مي شود، كوچكترين عدد صحيح m تعريف مي شود به طوري كه يك گراف G با m يال وجود داشته باشد كه در هر رنگ آميزي از يالهاي گراف G با r رنگ، يك كپي تكرنگ از گراف F وجود داشته باشد. كريولويچ و به طور جداگانه، دودك و پرالات براي مسير نشان داده اند كه براي n به اندازه كافي بزرگ𝑟̂(𝑃𝑛 ‚ 𝑟) ≤ 600𝑟2 ln 𝑟 𝑛 در اين مقاله با اثباتي كاملا متفاوت اين كران بالا را بهبود داده و ثابت مي كنيم 𝑟̂(𝑃𝑛 ‚ 𝑟) ≤ 18(1 + 𝑜𝑟(1))𝑟2 ln 𝑟 𝑛‚ لازم به ذكر است كه كران بالاي به دست آمده تقريبا بهينه است.

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].