An Investigation Of Some Problems In Ramsey Theory

This dissertation investigates the following three problems in Ramsey theory.(1) On the base of the asymptotic bounds for r(Cm,Kn) which were given by Caro, Li, Rousseau and Zhang , by analytic method and using the functionx ≥ 0; m ≥ 1, we give the asymptotic upper bounds for r(Wm,Kn) , i.e as n→∞,for fixed even m ≥ 4 andfor fixed odd m ≥ 5 , where C1 = C1(m) ≥ 0 and C2 = C2(m) ≥ 0. By the same method, we obtain the asymptotic relation between r(Wm,Kn) and r{Cm,Kn). Combining with the Spencer's asymptotic lower bounds for Ramsey functions we obtain the asymptotic bounds for r(K4, Kn). Moreover, we give the asymptotic upper bounds for r(Kk + Cm, Kn) , i.e as n→∞,for fixed even m ≥ 4 andfor fixed odd m ≥ 3, by employing mathematical induction on k, with the upper bounds for r(Wm, Kn) verified first.(2) Li has proved a Ramsey goodness result for graphs with many pendent edges. Inspired by this, by transplanting Li's method and mathematical induction on x(G), we give a Ramsey goodness result for graphs with a large pendent tree, i.e let H be any connected graph of ordern and let Hj be a graph of order n + j which is obtained from H by adding a pendent j - tree , and let Hj denote the class of all graphs: Hj, if j is large enough, the Ramsey nunber r(G,Hj) satisfies r(G, Hj) = (x - l)(n + j - 1) + s.(3) Zhou has proved that r(Bm,Wn) = 2n + 1 for m≥ l,n ≥ 5m + 3 and r(Bm,K2 + Cn) = 2n+3 forn ≥ 9 if m = 1 or n ≥ (m-l)(16m3+16m2-24rn-10)+1 if m ≥ 2. where the book Bm is the join K2 and KCm, Wn denotes a wheel with n spokes. Gu has given r(B1, K1 +Tn) = 2n +1 for n ≥ 3 and r(Bm, K1 + Tn) = 2n +1 for m ≥ 1,n ≥ 5m + 2. The above two are also Ramsey goodness result. Inspired by these, we obtain K2 + Tn is K3 - good by combinatorical method and mathematical inductoin on n with r(K3, K2 + T4) = 11 verified first.