Zongjian Sun

As recorded in S. P. Radiszowski, Small Ramsey numbers, Electronic. J. Comb., DS1 10,(2006),1-48, known results on the Ramsey numbers R(3;q) are
$R(3,25)\geq143, R(3,26)\geq150, R(3,28)\geq164, R(3,29)\geq174$.
New lower bounds for seven classical Ramsey numbers are obtained by considering some circulant graphs $G_{n}(A_{i})$ with $n\geq91$ whose orders might be either prime or not. The results are
Theorem. $R(3,24)\geq143,R(3,25)\geq153,R(3,26)\geq159,R(3,27)\geq167,R(3,28)\geq172,R(3,29)\geq182,R(3,30)\geq187$.
There are three formulas in S. P. Radiszowski, Small Ramsey numbers, Electronic. J. Combin., DS1 10,(2006),1-48.

R(3,4k+1)\geq6R(3,k+1)-5 (1)
R(5,k)\geq4R(3,k-1)-3 (2)
R(3,k,l+1)\geq4R(k,l)-3 (3)
As a consequence of Theorem and the formulas (1) (2) (3), we obtain
Corollary 1.
R(3,93)\geq853, R(3,97)\geq913, R(3,101)\geq949,
R(3,105)\geq997, R(3,109)\geq1027, R(3,113)\geq1087, R(3,117)\geq1117.
Corollary 2.
R(5,25)\geq569, R(5,26)\geq609,
R(5,27)\geq633,
R(5,28)\geq665, R(5,29)\geq685, R(5,30)\geq725, R(5,31)\geq745.
Corollary 3.
R(3,3,25)\geq569, R(3,3,26)\geq609, R(3,3,27)\geq633,
R(3,3,28)\geq665, R(3,3,29)\geq685, R(3,3,30)\geq725, R(3,3,31)\geq745.
The above analysis enables us to develop the following algorithm.
Algorithm
1)Given integer $n\geq5$, let $m=[\frac{n}{2}]$. Given a 2-partition $S=S_{1}\bigcup S_{2}$ of $S=[1,m]$ where both $S_{1}$ and $S_{2}$ are nonempty. Let $q_{i}=|S_{i}|$ for $i=1,\,2$. Let $i=1$.
2)Set $A_{i}=\{x|x\in Z_{n},x\in S_{i}=\ \text {or}\ n-x \in S_{i}\}$. Sort $A_{i}$ lexicographically. Assume that $A_{i}=\{x_{1},x_{2},\ldots\}$. Set $[A_{i}]=1$, $j=1$.
3)For $x_{j}\in S_{i}$, find
d_{i}(x_{j})=\{y|y\in A_{i}, y>x_{j}-y\in A_{i}\}.
If $|d_{i}(x_{j})|=0$ go to 5).
4)Find the $A_{i}$-colored chain starting with $x_{j}\in
S_{i}$. If $l_{i}(x_{j})\geq[A_{i}]$, let $[A_{i}]=l_{i}(x_{j})+1$
and print out this chain.
5)Increase $j$ by 1. If $j< q_{i}$, go to 3).
6)Let $k_i=[A_i]+1$. Increase $i$ by 1. If $i=2$, go to 2).
7)Print out $R(k_{1}+1,k_{2}+1)\geq n+1$ and the algorithm is terminated.
Given $n$ and the set $S_{1}$, the algorithm gives the clique number $[A_{2}]$ of $G_{n}(A_{2})$ and the first clique of length $[A_{i}]$.
We also points out other lower bounds such as
R(3,17)\geq92, R(3,18)\geq98, R(3,19)\geq106,
R(3,20)\geq109, R(3,21)\geq122, R(3,22)\geq125, R(3,23)\geq136.
These results were obtained by Wang Qingxian and Wang Gongben in 1994 and recorded in S. P. Radiszowski, Small Ramsey numbers, Electronic J. Combin., DS1 10,(2004),1-48. According to the reference in it, their paper [WWY1] has not been published. Hence our work verifies these results.