Does the Ramsey limit exist? What is the limit?
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For two graphs \( G\) and \( H\), let \( r(G,H)\) denote the smallest integer \( m\) satisfying the property that if the edges of the complete graph \( K_m\) are colored in red and blue, then there is either a subgraph isomorphic to \( G\) with all red edges or a subgraph isomorphic to \( H\) with all blue edges. The classical Ramsey numbers are those for the complete graphs and are denoted by \( r(s,t)= r(K_s, K_t)\). In the special case that \( s=t=n\), we simply write \( r(n)\) for \( r(n,n)\), and we call this the Ramsey number for \( K_n\).
The problem of accurately estimating \( r(n)\) is a notoriously difficult problem in combinatorics. The only known values [1] are \( r(3)=6\) and \( r(4)=18\).
For \( r(5)\), the best bounds [2][3] are \( 43 \leq r(5) \leq 49\). For the general \( r(n)\), the earliest bounds were:
\begin{equation} \frac{1}{e \sqrt 2} n 2^{n/2} < r(n) \leq \binom {2n2}{n1}. \end{equation}The upper bound follows from the fact that the Ramsey number \( r(k,l)\) satisfies \begin{equation} r(k,l) \leq r(k1,l)+r(k,l1) \end{equation}
with strict inequality if both \( r(k1,l)\) and \( r(k,l1)\) are even. To see this, if \( n = r(k1,l)+r(k,l1)\), for any vertex \( v\), there are either at least \( r(k1,l)\) red edges or at least \( r(k,l1)\) blue edges incident to \( v\). Therefore, there is either a red copy of \( K_k\) or a blue copy of \( K_l\). The strict inequality condition is a consequence of the fact that a graph on an odd number of vertices can not have all odd degrees.
The lower bound is established by a counting argument given by Erdős[4], which can be described as follows:
There are \( 2^{\binom{m}{2}}\) ways to color the edges of \( K_m\) in two colors. The number of colorings that contain a monochromatic \( K_n\) is at most
Therefore, there exists a coloring containing no monochromatic \( K_k\) if
This is true when
Very little progress has occurred in the intervening sixty years in improving these bounds. The best current bounds are
\begin{equation} (1+o(1)) \frac{ \sqrt 2} {e} n 2^{n/2} < r(n) < n^{c\log n/\log \log n}\binom {2n2}{n1}. \end{equation}
The upper bound is due to Conlon[5] and the lower bound is due to Spencer [6] by using the Lovász local lemma. Using these bounds, we see that \( r(n)^{1/n}\) is between \( \sqrt{2}\) and \( 4\).
Conjecture $100 (1947)
The limit \begin{equation*} \lim_{n \rightarrow \infty} r(n)^{1/n} \end{equation*} exists.
Problem $250 (1947)
Determine the value of \begin{equation*} c:=\lim_{n\rightarrow\infty} r(n)^{1/n} \end{equation*} if it exists.
Bibliography  

1 
R. E. Greenwood and A. M. Gleason,
Combinatorial relations and chromatic graphs,
Canad. J. Math. 7 (1955), 17.

2 
G. Exoo, A lower bound for \( R(5,5)\),
J. Graph Theory 13 (1989), 9798.

3 
B. D. McKay and S. P. Radziszowski,
Subgraph counting identities and Ramsey numbers,
J. Comb. Theory (B), 69 (1997), 193209.

4 
P. Erdös, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947), 292294.

5 
D. Conlon, A new upper bound for diagonal Ramsey numbers, Anals of Mathematics
170 (2009), 941960.

6 
J. Spencer, Ramsey's theorema new lower bound,
J. Comb. Theory Ser. A 18 (1975), 108115.
