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1.
A method is put forward to establish the lower bounds for somen-color classical Ramsey numbers
. With this method six new explicit lower boundsR
4(4) ≥458,R
3(5) ≥ 242,R
3(6)≥1070,R
3(7) ≥ 1214,R
3(8) ≥2834 andR
3(9) ≥ 5282 are obtained using a computer.
Project supported by Guangxi Natural Science Foundation 相似文献
2.
Ralph Faudree Ronald J. Gould Michael S. Jacobson Linda Lesniak 《Journal of Graph Theory》1990,14(6):723-730
For any graph G, let i(G) and μ;(G) denote the smallest number of vertices in a maximal independent set and maximal clique, respectively. For positive integers m and n, the lower Ramsey number s(m, n) is the largest integer p so that every graph of order p has i(G) ≤ m or μ;(G) ≤ n. In this paper we give several new lower bounds for s (m, n) as well as determine precisely the values s(1, n). 相似文献
3.
A new construction of self-complementary graphs containing no K10 or K11 is described. This construction gives the Ramsey number lower bounds r(10,10) ≧ 458 and r(11,11) ≧ 542. 相似文献
4.
Graph G is a (k, p)‐graph if G does not contain a complete graph on k vertices Kk, nor an independent set of order p. Given a (k, p)‐graph G and a (k, q)‐graph H, such that G and H contain an induced subgraph isomorphic to some Kk?1‐free graph M, we construct a (k, p + q ? 1)‐graph on n(G) + n(H) + n(M) vertices. This implies that R (k, p + q ? 1) ≥ R (k, p) + R (k, q) + n(M) ? 1, where R (s, t) is the classical two‐color Ramsey number. By applying this construction, and some its generalizations, we improve on 22 lower bounds for R (s, t), for various specific values of s and t. In particular, we obtain the following new lower bounds: R (4, 15) ≥ 153, R (6, 7) ≥ 111, R (6, 11) ≥ 253, R (7, 12) ≥ 416, and R (8, 13) ≥ 635. Most of the results did not require any use of computer algorithms. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 231–239, 2004 相似文献
5.
We describe an explicit construction whicy, for some fixed absolute positive constant ε, produces, for every integers>1 and all sufficiently largem, a graph on at least
vertices containing neither a clique of sizes nor an independent set of sizem.
Part of this work was done at the Institute for Advanced Study, Princeton, NJ 08540, USA. Research supported in part by a
USA Israeli BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry
at Tel Aviv University.
Research supported in part by a grant A1019901 of the Academy of Sciences of the Czech Republic and by a cooperative research
grant INT-9600919/ME-103 from the NSF (USA) and the MŠMT (Czech Republic). 相似文献
6.
This paper is a survey of the methods used for determining exact values and bounds for the classical Ramsey numbers in the case that the sets being colored are two-element sets. Results concerning the asymptotic behavior of the Ramsey functions R(k,l) and Rm(k) are also given. 相似文献
8.
Using methods developed by Graver and Yackel, and various computer algorithms, we show that 28 ≤ R(3, 8) ≤ 29, and R(3, 9) = 36, where R(k, l) is the classical Ramsey number for 2-coloring the edges of a complete graph. 相似文献
9.
The Ramsey number R(G1,G2,…,Gk) is the least integer p so that for any k-edge coloring of the complete graph Kp, there is a monochromatic copy of Gi of color i. In this paper, we derive upper bounds of R(G1,G2,…,Gk) for certain graphs Gi. In particular, these bounds show that R(9,9)6588 and R(10,10)23556 improving the previous best bounds of 6625 and 23854. 相似文献
10.
A coloring of the edges of a graph is called alocal k-coloring if every vertex is incident to edges of at mostk distinct colors. For a given graphG, thelocal Ramsey number, r
loc
k
(G), is the smallest integern such that any localk-coloring ofK
n
, (the complete graph onn vertices), contains a monochromatic copy ofG. The following conjecture of Gyárfás et al. is proved here: for each positive integerk there exists a constantc = c(k) such thatr
loc
k
(G) cr
k
(G), for every connected grraphG (wherer
k
(G) is theusual Ramsey number fork colors). Possible generalizations for hypergraphs are considered.On leave from the Institute of Mathematics, Technical University of Warsaw, Poland.While on leave at University of Louisville, Fall 1985. 相似文献
11.
Honghui Wan 《Journal of Graph Theory》1997,26(3):119-122
In this paper we show that for n ≥ 4, R(3, 3, ⋖, 3) < + 1. Consequently, a new bound for Schur numbers is also given. Also, for even n ≥ 6, the Schur number Sn is bounded by Sn < - n + 2. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 119–122, 1997 相似文献
12.
The irredundant Ramsey number s(m, n) is the smallest N such that in every red-blue coloring of the edges of KN, either the blue graph contains an m-element irredundant set or the red graph contains an n-element irredundant set. The definition of the mixed Ramsey number t(m, n) differs from s(m, n) in that the n-element irredundant set is replaced by an n-element independent set. We prove asymptotic lower bounds for s(n, n) and t(m, n) (with m fixed and n large) and a general upper bound for t(3, n). © 1993 John Wiley & Sons, Inc. 相似文献
13.
We investigate several bounds for both K2,m−K1,n Ramsey numbers and K2,m−K1,n bipartite Ramsey numbers, extending some previous results. Constructions based on certain geometric structures (designs, projective planes, unitals) yield classes of near-optimal bounds or even exact values. Moreover, relationships between these numbers are also discussed. 相似文献
14.
15.
Andrey B. Kupavskii Andrei M. Raigorodskii Maria V. Titova 《Discrete Mathematics》2013,313(22):2566-2574
In this paper we study the distance Ramsey number . The distance Ramsey number is the minimum number such that for any graph on vertices, either contains an induced -vertex subgraph isomorphic to a distance graph in or contains an induced -vertex subgraph isomorphic to the distance graph in . We obtain the upper and lower bounds on , which are similar to the bounds for the classical Ramsey number . 相似文献
16.
The Ramsey number r(H,Kn) is the smallest integer N so that each graph on N vertices that fails to contain H as a subgraph has independence number at least n. It is shown that r(K2,m,Kn)(m−1+o(1))(n/log n)2 and r(C2m,Kn)c(n/log n)m/(m−1) for m fixed and n→∞. Also r(K2,n,Kn)=Θ(n3/log2 n) and
. 相似文献
17.
18.
Michael Braun 《组合设计杂志》2019,27(11):682-687
An ‐arc in is a set of points such that each line contains at most of the selected points. It is well known that ‐arcs in correspond to projective linear codes. Let denote the maximal number of points for which an ‐arc in exists. In this paper we obtain improved lower bounds on by explicitly constructing ‐arcs. Some of the constructed ‐arcs correspond to linear codes meeting the Griesmer bound. All results are obtained by integer linear programming. 相似文献
19.
For given graphs , , the -color Ramsey number, denoted by , is the smallest integer such that if we arbitrarily color the edges of a complete graph of order with colors, then it always contains a monochromatic copy of colored with , for some . Let be a cycle of length and a star of order . In this paper, firstly we give a general upper bound of . In particular, for the 3-color case, we have and this bound is tight in some sense. Furthermore, we prove that for all and , and if is a prime power, then the equality holds. 相似文献
20.