A published paper or arXiv preprint answers the main question of Erdős Problem #1097 (theorem erdos_1097 in formal-conjectures) in full (proof, disproof, or the exact value asked for) with no hypotheses beyond the problem's own.

The problem, as stated in formal-conjectures (Apache 2.0):

The main conjecture: for any finite set of integers A with |A| = n, the number of distinct common differences in three-term arithmetic progressions is O(n^{3/2}). This conjecture was resolved negatively by showing that the problem is exactly equivalent to Bourgain's sums-differences question [Bo99], which was introduced as an arithmetic path towards the Kakeya conjecture. Under this equivalence: - The greatest achievable exponent for this problem is equal to the smallest constant c achievable for Bourgain's sums-differences question: |A -_G B| \ll \max(|A|, |B|, |A +_G B|)^c - The O(n^{3/2}) prediction is disproved because the lower bound has been shown to satisfy c \ge 1.77898 (due to Zheng and AlphaEvolve [GGTW25], improving on Lemm [Le15]), which is strictly greater than 3/2 = 1.5. - The best known upper bound is c \le 11/6 \approx 1.833 (due to Katz and Tao [KaTa99]). - While the specific O(n^{3/2}) prediction is resolved negatively, the general question of determining the exact optimal exponent c remains open.

References formal-conjectures lists for this problem (a starting point, not evidence):

• [Bo99] Bourgain, J., On the dimension of {K}akeya sets and related maximal