The standard Gale-Ryser theorem is for the existence of a $(0,1)$-matrix given exact row sums $R = (r_1, \dots, r_K)$ and exact column sums $C = (c_1, \dots, c_M)$. What if we relax the column sums and want at most those $c_j$ instead of exactly those $c_j$. What are the then necessary and sufficient conditions. Are they the same (except for the total sum obviously, which disappears)?
I'm only familiar with Krause's algorithmic proof of the Gale-Ryser theorem and would like to work on that extension of Gale-Ryser without resorting to flows.
