For positive integers s and t, the Ramsey number R ( s , t ) is the smallest positive integer n such that every graph of order n contains either a clique of order s or an independent set of order t. The triangle-free process begins with an empty graph of order n, and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. It has been an important tool in studying the asymptotic lower bound for R ( 3 , t ) . Cyclic graphs are vertex-transitive. The symmetry of cyclic graphs makes it easier to compute their independent numbers than related general graphs. In this paper, the cyclic triangle-free process is studied. The sizes of the parameter sets and the independence numbers of the graphs obtained by the cyclic triangle-free process are studied. Lower bounds on R ( 3 , t ) for small t’s are computed, and R ( 3 , 35 ) ≥ 237 , R ( 3 , 36 ) ≥ 244 , R ( 3 , 37 ) ≥ 255 , R ( 3 , 38 ) ≥ 267 , etc. are obtained based on the graphs obtained by the cyclic triangle-free process. Finally, some problems on the cyclic triangle-free process and R ( 3 , t ) are proposed.

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Cyclic triangle-free process (CTFP) is the cyclic analog of the triangle-free process. It begins with an empty graph of order n and generates a cyclic graph of order n by iteratively adding parameters, chosen uniformly at random, subject to the constraint that no triangle is formed in the cyclic graph obtained, until no more parameters can be added. The structure of a cyclic triangle-free graph of the prime order is different from that of composite integer order. Cyclic graphs of prime order have better properties than those of composite number order, which enables generating cyclic triangle-free graphs more efficiently. In this paper, a novel approach to generating cyclic triangle-free graphs of prime order is proposed. Based on the cyclic graphs of prime order, obtained by the CTFP and its variant, many new lower bounds on R 3 , t are computed, including R 3,34 ≥ 230 , R 3,35 ≥ 242 , R 3,36 ≥ 252 , R 3,37 ≥ 264 , R 3,38 ≥ 272 . Our experimental results demonstrate that all those related best known lower bounds, except the bound on R 3,34 , are improved by 5 or more.

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