چكيده انگليسي :
Mono multi bipartite Ramsey numbers designs and matrices Ali Arshi a arshi@math iut ac ir 2016 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Gholam Reza Omidi romidi@cc iut ac ir Advisor Dr Maryam Shahsiah m shahsiah@cc iut ac ir 2010 MSC 05C15 53C42 Keywords ramsey number edge coloring bipartite gragh AbstractRamsey s result was later rediscovered by Erd s and Szekeres and this topic has since been studiedextensively in the literature The rst nontrivial and most extensively studied case occurs whend 2 For this value of d the coloring described in Ramsey s theorem may be viewed as a coloring ofthe edges of KN Bialostocki and Voxman may have been inspired by the de nition of H n by Erd sand Graham when they de ned for a given graph G the number RM G to be the smallest integerN such that if the edges of the complete graph KN are colored with any number of colors then forthe resulting coloring there is either a monochromatic or a rainbow copy of G They showed thatthis number exists if and only if G is acyclic Eroh extended this de nition when she de ned for twographs G1 and G2 without isolated vertices the rainbow ramsey number RR G1 G2 to be the leastpositive integer N such that if the edges of KN are colored with any number of colors the resultinggraph contains either a monochromatic copy of G1 or a rainbow copy of G2 RR G1 G2 exists if andonly if G1 is a star or G2 is a forest Hence the existence of RR G1 G2 requires that either G1 or G2be bipartite This observation and the notion of bipartite ramsey numbers suggests another ramseyconcept Given two bipartite graphs G1 and G2 without isolated vertices the bipartite rainbowramsey number BRR G1 G2 is the smallest integer N such that any edge coloring of KN N containsa monochromatic copy of G1 or a rainbow copy of G2 An edge coloring of a graph G is assigningcolors to the edges of G A graph with colored edges is said to be monochromatic if all its edges havethe same color and is said to be rainbow if no two of its edges have the same color For the various
