确定了集合交集为等差数列时的最大集合数，突破了经典猜想。
Exact values and exact upper bounds for families of integers with arithmetic progression intersections (Erdős Problem #272)
- 通过穷举计算精确求解3到12个元素时的最大集合数。
- 证明了在所有含公共元素的族中，上界由Szabo构造达到。
- 提出新结构定理，为解决核心猜想提供关键路径。
设t(N)为最大整数t，使得存在互异的集合A₁,…,Aₜ⊆{1,…,N}，满足任意i≠j时Aᵢ∩Aⱼ为非空等差数列（Erdős问题#272）。Simonovits与Sos证明t(N)=O(N²)，并猜想最大值为½×(N²−N)+1；Szabo通过构造给出t(N)≥½×(N²−N)+1+⌊(N−1)/4⌋，并证明t(N)=N²/2+O(N^{5/3}(log N)³)，同时提出两个开放问题：是否t(N)=½×(N²−N)+O(N)，以及极值族中是否存在共用元素（核问题）。本文通过穷举计算精确确定了3≤N≤12时的t(N)，发现均等于Szabo的下界。进一步证明，在所有含共同元素的族（星形族）中，该下界为严格最大。其证明结合了针对阶梯区域的“缺陷一”计数不等式和新结构定理：任何非等差成员必包含一个坏对，其他成员无法共享。因此，更强猜想等价于仍需证明的核猜想——极值族必有公共元素。本文还给出了非星形极值族的初步结构约束。
原文摘要 · Abstract (English)
Let $t(N)$ be the largest $t$ for which there exist distinct sets $A_1,\dots,A_t \subseteq \{1,\dots,N\}$ such that $A_i \cap A_j$ is a nonempty arithmetic progression for all $i \neq j$ (Erdos Problem #272). Simonovits and Sos proved $t(N)=O(N^2)$ and conjectured $\binom{N}{2}+1$ is best possible; Szabo disproved this by a construction giving $t(N) \geq \binom{N}{2}+1+\lfloor(N-1)/4\rfloor$, proved the asymptotics $t(N)=N^2/2+O(N^{5/3}(\log N)^3)$, and asked whether $t(N)=\binom{N}{2}+O(N)$ and whether some element lies in all sets of any extremal family (the kernel question). We determine $t(N)$ exactly for all $3 \leq N \leq 12$ by exhaustive computation: in this entire range Szabo's lower bound is exact, and we conjecture that $t(N)=\binom{N}{2}+1+\lfloor(N-1)/4\rfloor$ for every $N$. Towards the matching upper bound we prove, for every $N$, that Szabo's bound is the exact maximum over all families with a common element (starred families). The proof combines a self-contained ``defect-one'' counting inequality for staircase regions with a new structural theorem: every non-progression member of such a family contains a bad pair that no other member can share. Consequently the sharpened conjecture reduces to a single remaining statement, namely Szabo's kernel conjecture that some element lies in all sets of an extremal family, and we prove first structural constraints on putative non-starred extremal families.
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