## “School of Mathematics”

Back to Papers Home
Back to Papers of School of Mathematics

Paper   IPM / M / 16310
 School of Mathematics Title: On the multicolor size Ramsey number of stars and cliques Author(s): Maryam Shahsiah (Joint with M. Miralaei) Status: Published Journal: Discrete Math. Vol.: 343 Year: 2020 Pages: 111899 Supported by: IPM
Abstract:
For given graphs H1â, â...â, âHt â, âwe say that G is Ramsey forâ â H1â, â...â, âHt and we writeâ â G→ (H1â, â...â, âHt) â, âif no matter how one colors the edges ofâ â G with t colorsâ, âsay 1,...â, ât â, âthere exists a monochromatic copy of Hi in the i th colorâ â for some 1 ≤ it â. âThe multicolor Ramsey numberâ â r(H1â, â...â, âHt) is the smallest integer n such that the complete graphâ â Kn is Ramsey for (H1â, â...â, âHt) â. â The multicolor size Ramsey numberâ â r(H1â, â...â, âHt) is defined asâ â min{ |E(G)|â: âG→ (H1â, â...â, âHt) } â, âwhile the restricted size Ramsey numberâ â r*(H1â, â...â, âHt) is defined asâ â
 ââ ^ r * (H1â, â...â, âHt)= min { |E(G)|â: âG→ (H1â, â...â, âHt)â â and â â|V(G)|=r(H1â, â...â, âHt) }â.â
â â â In 1978 Erdös et alâ. âinitiated the study of the size Ramsey numbers ofâ â graphsâ. âAfterwardsâ, âsize Ramsey numbersâ â have been studied with particular focus on the case of treesâ, âsparse graphs and bounded degree graphsâ. â In this paperâ, âthe order of magnitude ofâ â multicolor size Ramsey number of stars and cliques is determined in terms of r(K1,q1â, â...â, âK1,qn) and r(Kp1â, â...â, âKpm) â. â Moreoverâ, âin some special casesâ, â restricted size Ramsey number of stars and cliques is determined exactlyâ. âOur results haveâ, âup to a constant factorâ, âsimilar order of magnitude andâ â generalize significantly a well known result of Faudree and Sheehan.
â