Glock–Kühn–Lo–Osthus high-girth Steiner system conjecture

For integers $q>r\ge2$, an $(n,q,r)$-Steiner system is a collection of $q$-element subsets of an $n$-element set in which every $r$-element subset lies in exactly one block. A $(j,i)$-configuration is a set of $i$ blocks spanning at most $j$ vertices; $n$ is admissible when

$$\binom{q-i}{r-i}\mid\binom{n-i}{r-i}$$

for all $0\le i\le r-1$. Glock–Kühn–Lo–Osthus's conjecture. For all integers $q>r\ge2$ and every integer $g\ge2$, there exists $n_0$ such that for all admissible $n\ge n_0$, there exists an $(n,q,r)$-Steiner system with no $(i(q-r)+r,i)$-configurations for all $2\le i\le g$. The text presents this as a common generalization of Erdős's conjecture and the Existence Conjecture; its resolution is not stated in the supplied passage.