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.
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.
The planar Ramsey numbers for all pairs of complete graphs was determined in Steinberg and Tovey (1993).
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.
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.
One can also consider an asymmetric version of the on-line Ramsey number.
Combinatorics receives a full chapter treatment that extends beyond the combinations and permutations material by delving into non-standard topics such as Latin squares, finite projective planes, balanced incomplete block designs, coding theory, partitions, occupancy problems, Stirling numbers, Ramsey numbers
, and systems of distinct representatives.