Ledley has been voted by fashion magazine Vogue as the eighth best-dressed player at the Euros, with old pal and team-mate Aaron Ramsey number
Given two graphs G and H, the Ramsey number R(G, H) is the smallest integer n such that every graph F on n vertices contains a copy of G, or its complement [bar.F] contains a copy of H.
For two given graphs G and H the planar Ramsey number PR(G, H) is the smallest integer n such that every planar graph F on n vertices either contains a copy of G, or its complement contains a copy of H.
Topics include Ramsey number
theory (that there cannot be complete disorder and in any large system there must always be some structure), additive number theory, multiplicative number theory, combinatorial games, sequences, elementary number theory and graph theory.
The resulting minimum number--which equals 6 when x is 3 and y is 3--is called a Ramsey number.
"This was the smallest unsolved Ramsey number," Radziszowski says.
For any natural numbers m and n, the (classical) Ramsey number r = r(m, n) is the smallest natural number r such that, for any red-blue edge colouring (R, B) of the complete graph [K.sub.r] on r vertices, it holds that either [K.sub.m] [subset or equal to] R or [K.sub.n] [subset or equal to] B, or perhaps both.
The notion of a Ramsey number may also be defined in terms of independent sets in graphs.
This notion was first studied by Erdos, Faudree, Rousseau, and Schelp (8), as a variation on the usual Ramsey number
r(H) (which is the least n such that [K.sub.n] [right arrow] H, where [K.sub.n] is a clique on n vertices).
It discusses basic definitions and notations, bi-color diagonal classical Ramsey numbers
, Paley graphs and lower bounds for R(k, k), bi-color off-diagonal and multicolor classical Ramsey numbers
, generalized Ramsey numbers
, Folkman numbers, the Erd|s-Hajnal conjecture, other Ramsey-type problems in graph theory, der Waerden numbers and Szemeredi's theorem, Sidon-Ramsey numbers, games in Ramsey theory, local Ramsey theory, set-coloring Ramsey theory, and other problems and conjectures.