  Tverberg-type theory aims to establish sufficient conditions for a simplicial
complex $\Sigma$ such that every continuous map $f\colon \Sigma \to
\mathbb{R}^d$ maps $q$ points from pairwise disjoint faces to the same point in
$\mathbb{R}^d$. Such results are plentiful for $q$ a power of a prime. However,
for $q$ with at least two distinct prime divisors, results that guarantee the
existence of $q$-fold points of coincidence are non-existent -- aside from
immediate corollaries of the prime power case. Here we present a general method
that yields such results beyond the case of prime powers. In particular, we
prove previously conjectured upper bounds for the topological Tverberg problem
for all $q$.
