  In 1973, Erd\H{o}s conjectured the existence of high girth $(n,3,2)$-Steiner
systems. Recently, Glock, K\"{u}hn, Lo, and Osthus and independently Bohman and
Warnke proved the approximate version of Erd\H{o}s' conjecture. Just this year,
Kwan, Sah, Sawhney, and Simkin proved Erd\H{o}s' conjecture. As for Steiner
systems with more general parameters, Glock, K\"{u}hn, Lo, and Osthus
conjectured the existence of high girth $(n,q,r)$-Steiner systems. We prove the
approximate version of their conjecture.
  This result follows from our general main results which concern finding
perfect or almost perfect matchings in a hypergraph $G$ avoiding a given set of
submatchings (which we view as a hypergraph $H$ where $V(H)=E(G)$). Our first
main result is a common generalization of the classical theorems of Pippenger
(for finding an almost perfect matching) and Ajtai, Koml\'os, Pintz, Spencer,
and Szemer\'edi (for finding an independent set in girth five hypergraphs).
More generally, we prove this for coloring and even list coloring, and also
generalize this further to when $H$ is a hypergraph with small codegrees (for
which high girth designs is a specific instance). Indeed, the coloring version
of our result even yields an almost partition of $K_n^r$ into approximate high
girth $(n,q,r)$-Steiner systems.
  Our main results also imply the existence of a perfect matching in a
bipartite hypergraph where the parts have slightly unbalanced degrees. This has
a number of applications; for example, it proves the existence of $\Delta$
pairwise disjoint list colorings in the setting of Kahn's theorem; it also
proves asymptotic versions of various rainbow matching results in the sparse
setting (where the number of times a color appears could be much smaller than
the number of colors) and even the existence of many pairwise disjoint rainbow
matchings in such circumstances.
