  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. In this paper, we study the lower bounds for small Ramsey numbers
on hypergraphs by constructing counter-examples 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 $r_5(6,6) \ge 72$, where there is
only trivial lower bound on $5$-hypergraphs 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 $k \ge 4$.
