On Some Small Classical Ramsey Numbers
نویسندگان
چکیده
منابع مشابه
On Some Small Classical Ramsey Numbers
This note is a report on a computer investigation of some small classical Ramsey numbers. We establish new lower bounds for the classical Ramsey numbers R(3, 11) and R(4, 8). In the first case, the bound is improved from 46 (a record that had stood for 46 years) to 47; and in the second case the bound is improved from 57 to 58. The classical Ramsey number R(s, t) is the smallest integer n such ...
متن کاملBounds on Some Ramsey Numbers
For graphs G1, G2, · · · , Gm, the Ramsey number R(G1, G2, · · · , Gm) is defined to be the smallest integer n such that anym-coloring of the edges of the complete graphKn must include a monochromatic Gi in color i, for some i. In this note we establish several lower and upper bounds for some Ramsey numbers involving quadrilateral C4, including R(C4,K9) ≤ 32, 19 ≤ R(C4, C4,K4) ≤ 22, 31 ≤ R(C4, ...
متن کاملOn Some Multicolor Ramsey Numbers
The Ramsey number R(G1, G2, G3) is the smallest positive integer n such that for all 3-colorings of the edges of Kn there is a monochromatic G1 in the first color, G2 in the second color, or G3 in the third color. We study the bounds on various 3-color Ramsey numbers R(G1, G2, G3), where Gi ∈ {K3,K3 + e,K4 − e,K4}. The minimal and maximal combinations of Gi’s correspond to the classical Ramsey ...
متن کاملSome recent results on Ramsey-type numbers
In this paper we survey author’s recent results on quantitative extensions of Ramsey theory. In particular, we discuss our recent results on Folkman numbers, induced bipartite Ramsey graphs, and explicit constructions of Ramsey graphs.
متن کاملOn Some Three-Color Ramsey Numbers
In this paper we study three-color Ramsey numbers. Let Ki,j denote a complete i by j bipartite graph. We shall show that (i) for any connected graphs G1, G2 and G3, if r(G1, G2) ≥ s(G3), then r(G1, G2, G3) ≥ (r(G1, G2) − 1)(χ(G3) − 1) + s(G3), where s(G3) is the chromatic surplus of G3; (ii)(k + m − 2)(n − 1) + 1 ≤ r(K1,k,K1,m,Kn) ≤ (k + m − 1)(n − 1) + 1, and if k or m is odd, the second inequ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2013
ISSN: 1077-8926
DOI: 10.37236/3137