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 graph $K_N$,there must exist either an all-red $K_a$ or an all-blue $K_b$.

Frank Ramsey proved these numbers always exist.He famously pointed out that among any 6 people, some three are mutual friendsor some three mutual non-friends. That is, $R(3,3)\le 6$.Since a red pentagon with a blue pentagram drawn inside it has no monochromatic triangle,$R(3,3)\ge 6$. 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-colorings of edges, etc (any number ofarguments), and allow general sets of graphs, not just pairs of complete ones.

Ramsey numbers are very difficult to determine.To prove lower bounds, construct good edge-colorings of some $K_N$ and, use a clique-finderto find the largest mono-colored cliques.To prove upper bounds, the main tool has been $R(a,b)\le R(a-1,b)+R(b-1,a)$which implies $R(a,b)\le \left(\genfrac{}{}{0pt}{}{a+b-2}{a-1}\right)$ and then $R(k)\le [1+o(1)]4^k/(4\sqrt{\pi k}$when $k\to \mathrm{\infty }$.From considering random colorings and using a probabilistic nonconstructive existenceargument, one may show $R(k)\ge k{2}^{k/2}[o(1)+\sqrt{2}/e]$.It is known that $R(1)=1$, $R(2)=2$, $R(3)=6$, $R(4)=18$, and $43\le R(5)\le 49$.For a survey of the best upper and lower bounds available on smallRamsey numbers, seehttp://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 $\overrightarrow{R}(n)$ denote the least integer $N$ so that any tournament (complete directed graphwith singly-directed arcs) with $\ge N$ vertices contains an acyclic (also called “transitive”)$n$-node tournament. (Analogies: 2-color the edges $\to$ two directions for arcs.Monochromatic $\to$ acyclic, i.e. all arcs “point one way.”)

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