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For any $g\geq 2$, if $n$ is sufficiently large and $\equiv 1,3\pmod{6}$ then there exists a 3-uniform hypergraph on $n$ vertices such that
every pair of vertices is contained in exactly one edge (i.e. the graph is a Steiner triple system) and
for any $2\leq j\leq g$ any collection of $j$ edges contains at least $j+3$ vertices.
#207
:
[Er76]
combinatorics
,
hypergraphs
Proved by Kwan, Sah, Sawhney, and Simkin
[KSSS22b]
.
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