  Erd\H{o}s and Szekeres showed in 1978 that for any four positive integers
satisfying m_1+m_2 = n_1+n_2, the two binomial coefficients (m_1+m_2)!/m_1!
m_2! and (n_1+n_2)!/n_1! n_2! have a common divisor >1. The analogous statement
for families of k k-nomial coefficients (k>1) was conjectured in 1997 by David
Wasserman.
  Erd\H{o}s and Szekeres remark that if m_1, m_2, n_1, n_2 as above are all >1,
there is probably a lower bound on the common divisor in question which goes to
infinity as a function of m_1+m_2. Such a bound is here obtained.
  Results are proved that narrow the class of possible counterexamples to
Wasserman's conjecture. On the other hand, several plausible generalizations of
that conjecture are shown to be false.
