Lower bound for \( r(4, n) \)
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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 offdiagonal Ramsey numbers (that is, where \( s\neq t\), the known values are \( r(3,4)= 9\), \( r(3,5)=14\), \( r(3,6)=18\), \( r(3,7)=23\), \( r(3,8)=28\), \( r(3,9)=36\) and \( r(4,5)=25\) while \( 35 \leq r(4,6) \leq 41\) (see the dynamic survey of Radzisowski on small Ramsey numbers in the Electronic Journal of Combinatorics for more bounds and references).
For \( k=3\), Kim [5] recently proved a new lower bound which matches the previous upper bound for \( r(3,n)\) (up to a constant factor), so it is now known that
\begin{equation} \frac{c n^2}{\log n} < r(3,n) < (1+o(1)) \frac{ n^2}{\log n}. \end{equation}Ajtai, Komlós and Szemerédi [3] earlier gave the upper bound of \( c' \frac{ n^2}{\log n}\) and Shearer [6], [7] replaced \( c'\) by \( 1+o(1)\). It would be of interest to have an asymptotic formula for \( r(3,n)\).
The best known constructive lower bound for \( r(3,n)\) is due to Alon [1]:
For \( r(4,n)\), the best lower bound known is \( c (n \log n)^{5/2}\) due to Spencer [8], by using the Lovász local lemma. The best upper bound known is \( c' n^3 / \log ^2 n\), proved by Ajtai, Komlós and Szemerédi [3]. So there is a nontrivial gap still remaining, as repeatedly pointed out in many problems papers [9] of Erdös.
Problem [9]
Prove or disprove that \begin{equation} r(4,n) > \frac{n^3}{\log ^c n} \end{equation} for some \(c\), provided \(n\) is sufficiently large.
Bibliography  

1  N. Alon. Explicit Ramsey graphs and orthonormal labelings, Elec. J. Comb. 1 (1994), R12 (8pp)

2  F. R. K. Chung, R. Cleve and P. Dagum. A note on constructive lower bounds for the Ramsey numbers \( R(3,t).\) J. Comb. Theory, Ser. B 57 (1993), 150155.

3 
M. Ajtai, J. Komlós and E. Szemerédi, A note on
Ramsey numbers, J. Comb. Theory Ser. A 29 (1980),
354360.

4  P. Erdös, On the construction of certain graphs, J. Comb. Theory 17 (1966), 149153.

5 
J. H. Kim,
The Ramsey number \( R(3,t)\) has order of magnitude
\( t^2/ \log t\),
Random Structures and Algorithms 7 (1995), 173207.

6 
J. Shearer,
A note on the independence number of trianglefree graphs,
Discrete Math. 46 (1983), 8387.

7 
J. Shearer,
A note on the independence number of trianglefree graphs II,
J. Comb. Theory (B) 53 (1991), 300307.

8 
J. Spencer, Asymptotic lower bounds for Ramsey functions,
Discrete Math. 20 (1977/78), 6976.

9 
P. Erdös,
Problems and results on graphs and hypergraphs: similarities and
differences,
Mathematics of Ramsey theory, Algorithms Combin., 5,
(J. Nešetril and V. Rödl, eds.),
1228,
Springer, Berlin, 1990.
