For every n, the maximum size of a 3-uniform set system on [n] with VC-dimension 2 is C(n,3) for n≤5, 13 for n=6, and C(n-1,2)+1 for n≥7.
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Largest $3$-uniform set systems with VC-dimension $2$
For every n, the maximum size of a 3-uniform set system on [n] with VC-dimension 2 is C(n,3) for n≤5, 13 for n=6, and C(n-1,2)+1 for n≥7.