Ramsey numbers

Define R(a,b) to be the least integer N such that, in any red-blue 2-coloring of the edges of a N-vertex complete graphMathworldPlanetmath KN, there must exist either an all-red Ka or an all-blue Kb.

Frank Ramsey proved these numbers always exist. He famously pointed out that among any 6 people, some three are mutual friends or some three mutual non-friends. That is, R(3,3)6. Since a red pentagon with a blue pentagram drawn inside it has no monochromatic triangle, R(3,3)6. So R(3,3)=6.

Special attention is usually paid to the diagonal R(k,k), which is often just written R(k).

One can also generalize this in various ways, e.g. consider R(a,b,c) for three-coloringsMathworldPlanetmath of edges, etc (any number of argumentsMathworldPlanetmath), and allow general sets of graphs, not just pairs of completePlanetmathPlanetmathPlanetmathPlanetmath ones.

Ramsey numbersMathworldPlanetmath are very difficult to determine. To prove lower bounds, construct good edge-colorings of some KN and, use a clique-finder to find the largest mono-colored cliques. To prove upper bounds, the main tool has been R(a,b)R(a-1,b)+R(b-1,a) which implies R(a,b)(a+b-2a-1) and then R(k)[1+o(1)]4k/(4πk when k. From considering random colorings and using a probabilistic nonconstructive existence argument, one may show R(k)k2k/2[o(1)+2/e]. It is known that R(1)=1, R(2)=2, R(3)=6, R(4)=18, and 43R(5)49. For a survey of the best upper and lower bounds available on small Ramsey numbers, see http://www.combinatorics.org/Surveys/ds1.pdfRadziszowski’s survey (http://www.cs.rit.edu/ spr/alternate link). Another kind of Ramsey-like number which has not gotten as much attention as it deserves, are Ramsey numbers for directed graphs. Let R(n) denote the least integer N so that any tournamentMathworldPlanetmath (complete directed graphMathworldPlanetmath with singly-directed arcs) with N vertices contains an acyclic (also called “transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath”) n-node tournament. (AnalogiesMathworldPlanetmath: 2-color the edges two directions for arcs. Monochromatic acyclic, i.e. all arcs “point one way.”)

Again, to prove lower bounds, construct good tournaments and apply something like a clique-finder (but instead aimed at trying to find the largest acyclic induced subgraphMathworldPlanetmath). To prove upper bounds, the main tool is R(n+1)2R(n). That can be used to show the upper bound, and random-orientation arguments combined with a nonconstructive probabilistic existence argument show the lower bound, in [1+o(1)]2n+1)/2R(n)552n-7. It is known that R(1)=1, R(2)=2, R(3)=4, R(4)=8, R(5)=14, R(6)=28, and 32R(7)55. For a full survey of directed graph Ramsey numbers includng proofs and refererences, see http://www.rangevoting.org/PuzzRamsey.htmlSmith’s survey.

Title Ramsey numbers
Canonical name RamseyNumbers
Date of creation 2013-03-22 16:16:32
Last modified on 2013-03-22 16:16:32
Owner wdsmith (13774)
Last modified by wdsmith (13774)
Numerical id 10
Author wdsmith (13774)
Entry type Definition
Classification msc 05D10
Related topic RamseysTheorem2
Defines Ramsey numbers