The Ramsey number r_k(p,q) is the smallest integer N that satisfies for every red-blue coloring on k-subsets of [N], there exist p integers such that any k-subset of them is red, or q integers such that any k-subset of them is blue, fn this paper, we study the lower bounds for small Ramsey numbers on hypergraphs by constructing counterexamples and recurrence relations. We present a new algorithm to prove lower bounds for r_k(k + 1,k + 1). In particular, our algorithm is able to prove r_5(6,6) > 72, where there is no lower bound on 5-hypcrgraphs before this work. We also provide several recurrence relations to calculate lower bounds based on lower bound values on smaller p and q. Combining both of them, we achieve new lower bounds for r_k(p,q) on arbitrary p, q, and k ≥ 4.
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