Off‐diagonal and asymptotic results on the Ramsey number r(K2,m,K2,n)

An upper bound on the Ramsey number r(K_2,n‐s,K_2,n) where s ≥ 2 is presented. Considering certain r(K_2,n‐s,K_2,n)‐colorings obtained from strongly regular graphs, we additionally prove that this bound matches the exact value of r(K_2,n‐s,K_2,n) in infinitely many cases if holds. Moreover, the asymptotic behavior of r(K_2,m,K_2,n) is studied for n being sufficiently large depending on m. We conclude with a table of all known Ramsey numbers r(K_2,m,K_2,n) where m,n ≤ 10. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 252–268, 2003     

素数阶循环图和经典Ramsey数R（4，n）的三个新下界

研究了素数阶循环圈的基本性质，提出了寻求有效参数构造正则循环圈的新方法，得到了3个经典Ramsey数的新下界：R（4，17）≥164，R（4，18）≥182，R（4，22）≥282．这前2个结果填补了关于Ramsey数综述[2]的上下界表中的2个空白，第3个结果超过了目前已知的最好下界R（4，22）≥258，     

New bounds for the distance Ramsey number

In this paper we study the distance Ramsey number $R_D(s,t,d)$. The distance Ramsey number $R_D(s,t,d)$ is the minimum number $n$ such that for any graph $G$ on $n$ vertices, either $G$ contains an induced $s$-vertex subgraph isomorphic to a distance graph in ${\mathbb{R}}^d$ or $\stackrel{?}{G}$ contains an induced $t$-vertex subgraph isomorphic to the distance graph in ${\mathbb{R}}^d$. We obtain the upper and lower bounds on $R_D(s,s,d)$, which are similar to the bounds for the classical Ramsey number $R(?\frac{s}{[d/2]}?,?\frac{s}{[d/2]}?)$.     

5-chromatic strongly regular graphs

In this paper, we begin the determination of all primitive strongly regular graphs with chromatic number equal to 5. Using eigenvalue techniques, we show that there are at most 43 possible parameter sets for such a graph. For each parameter set, we must decide which strongly regular graphs, if any, possessing the set are 5-chromatic. In this way, we deal completely with 34 of these parameter sets using eigenvalue techniques and computer enumerations.     

Lifting constructions of strongly regular Cayley graphs

We give two “lifting” constructions of strongly regular Cayley graphs. In the first construction we “lift” a cyclotomic strongly regular graph by using a subdifference set of the Singer difference sets. The second construction uses quadratic forms over finite fields and it is a common generalization of the construction of the affine polar graphs [7] and a construction of strongly regular Cayley graphs given in [15]. The two constructions are related in the following way: the second construction can be viewed as a recursive construction, and the strongly regular Cayley graphs obtained from the first construction can serve as starters for the second construction. We also obtain association schemes from the second construction.     

The graph Ramsey number R(Fℓ,K6)

For a given pair of two graphs , let be the smallest positive integer such that for any graph of order , either contains as a subgraph or the complement of contains as a subgraph. Baskoro, Broersma and Surahmat (2005) conjectured that for , where is the join of and . In this paper, we prove that this conjecture is true for the case .     

Size Ramsey numbers for some regular graphs

P. Erdös, R.J. Faudree, C.C. Rousseau and R.H. Schelp [P. Erdös, R.J. Faudree, C.C. Rousseau, R.H. Schelp, The size Ramsey number, Period. Math. Hungar. 9 (1978) 145-161] studied the asymptotic behaviour of for certain graphs G,H. In this paper there will be given a lower bound for the diagonal size Ramsey number of K_n,n,n. The result is a generalization of a theorem for K_n,n given by P. Erdös and C.C. Rousseau [P. Erdös, C.C. Rousseau, The size Ramsey numbers of a complete bipartite graph, Discrete Math. 113 (1993) 259-262].Moreover, an open question for bounds for size Ramsey number of each n-regular graph of order n+t for t>n−1 is posed.     

Star-critical Ramsey numbers

The graph Ramsey numberR(G,H) is the smallest integer r such that every 2-coloring of the edges of K_r contains either a red copy of G or a blue copy of H. We find the largest star that can be removed from K_r such that the underlying graph is still forced to have a red G or a blue H. Thus, we introduce the star-critical Ramsey numberr_∗(G,H) as the smallest integer k such that every 2-coloring of the edges of K_r−K_1,r−1−k contains either a red copy of G or a blue copy of H. We find the star-critical Ramsey number for trees versus complete graphs, multiple copies of K₂ and K₃, and paths versus a 4-cycle. In addition to finding the star-critical Ramsey numbers, the critical graphs are classified for R(T_n,K_m), R(nK₂,mK₂) and R(P_n,C₄).     

Ramsey数R_m(3)的计算及Schur数的新上界

利用抽屉原理,给出了Ramsey数Rm(3)的一个递推公式,得到Rm(3)准确值计算的一个具体表达式,并利用Rm(3)的计算公式给出了Schur数的一个新的上界。