Lower bound for \(r(k, n)\)
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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 \( r(s,t)= r(K_s, K_t)\).
For general \( k\), the best asymptotic bounds for \( r(k,n)\), for \( n\) large, are as follows:
\begin{equation} c \left( \frac{n}{\log n} \right)^{(k+1)/2} < r(k,n) < (1+o(1)) \frac{n^{k-1}}{\log^{k-2} n}. \end{equation}The upper bound is a recent result of Li and Rousseau [2] who extend Shearer's method to improve the constant factor for the bounds in [1]. The lower bound is given in [3].
Conjecture (1947)
For fixed \(k\), \begin{equation} r(k,n) > \frac{n^{k-1}}{\log^c k} \end{equation} for a suitable constant \(c >0\) and \(n\) large.
Bibliography | |
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1 |
M. Ajtai, J. Komlós and E. Szemerédi, A note on
Ramsey numbers, J. Comb. Theory Ser. A 29 (1980),
354-360.
|
2 |
Y. Li and C. C. Rousseau,
Bounds for independence numbers and classical Ramsey numbers,
preprint.
|
3 |
J. Spencer, Asymptotic lower bounds for Ramsey functions,
Discrete Math. 20 (1977/78), 69-76.
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