| Given arbitrary graphs $Gsb1,Gsb2,...,Gsb{c}$ we say that a graph F "arrows" $Gsb1,Gsb2,...,Gsb{c}$ (in symbols $F to Gsb1,Gsb2,...,Gsb{c})$ if in every c-coloring of the edges of F there exists, for some i, 1 $le$ i $le$ c, a monochromatic copy of $Gsb{i}$ in color i.;If F = $Ksb{n}$ we define the Ramsey number $r(Gsb1,Gsb2,...,Gsb{c})$ to be the smallest $n in$ N such that $Ksb{n} to (Gsb1,Gsb2,...,Gsb{c}).$ Similarly, we define the bipartite Ramsey number $br(Gsb1,Gsb2,...,Gsb{c})$ to be the smallest $n in$ N such that $Ksb{n,n} to (Gsb1,Gsb2,...,Gsb{c}).$.;In their paper, "The Ramsey Number for Stripes", E. J. Cockayne and P. J. Lorimer define "stripes" to be multiple copies of $Psb2$ and they determine the Ramsey number $r(msb1Psb2,msb2Psb2,...,msb{c}Psb2)$ where c denotes the number of colors. In this thesis we consider the Ramsey number $r(msb1Psb{n},msb2Psb{n},...,msb{c}Psb{n})$, c, n $in$ N, and are able to establish the 3-color result for the case when n = 3 and a partial result for the case when n = 4. That is, we prove the following two theorems:;Theorem. If $msb1 ge msb2 ge msb3 ge 1, msb1 ge 2$ then $r(msb1Psb3, msb2Psb3, msb3Psb3) = 3msb1 + msb2 + msb3 - 2$.;Theorem. If $msb1 ge msb2 > msb3 ge 1,$ then $r(msb1Psb4, msb2Psb4, msb3Psb4) = 4msb1 + 2msb2 + 2msb3 - 2$.;Moreover, we are able to establish the 2-color bipartite Ramsey number for the case when n = 3.;Theorem. If $msb1 ge msb2 ge 1,msb1 ge 2,$ then $$br(msb1Psb3, msb2Psb3) = leftlfloor{3msb1 - msb2over2}rightrfloor + msb2.$$.;The remainder of this thesis gives a survey of results from classical and generalized Ramsey theory. |
