Prove the conjectured exact extremal value

Prove that, for sufficiently large n relative to d, the maximum size of a (d+1)-uniform set system on [n] with VC-dimension at most d equals \binom{n-1}{d}+\binom{n-4}{d-2}.

Background

Ahlswede and Khachatrian constructed a family attaining \binom{n-1}{d}+\binom{n-4}{d-2}, and Mubayi and Zhao subsequently conjectured that this construction is optimal. The paper’s main upper bound supports this conjectured value by removing the dominant term in the previous gap, but it falls short of proving exact optimality.

References

Later, Mubayi and Zhao showed that when d is a prime power and n is sufficiently large compared to d, the Frankl--Pach upper bound can be improved to \binom{n}{d} - \Omega_{d}(\log{n}). In a recent development, Ge, Zhao, and three of the authors showed that the Frankl--Pach upper bound can be improved to \binom{n}{d} - 1 for all positive integers n,d with d\ge 2 and n\geq 2d+2. These results, while having elegant proofs, reveal how challenging it has been to make meaningful advances.

— Uniform set systems with small VC-dimension  (2501.13850 - Chao et al., 23 Jan 2025) in Section 1, subsection “Background”