Pensieve header: Computing some Ramsey numbers.

R[_,1]=1;
R[1,_] = 1;
R[m_,n_] := R[m,n] = R[m-1,n]+R[m,n-1];
R[nn__, m_,n_] := R[nn,m,n] = R[nn, R[m,n]];

R[3,3]

R[4,4]

R[3,3,3]

R[4,4,4]

R[4,4,4,4]

$RecursionLimit=∞;
R[4,4,4,4]

R[5,5,5,5]