具有算术级数交集的整数族的精确值和精确上界（埃尔德什问题#272）
Exact values and exact upper bounds for families of integers with arithmetic progression intersections (Erdős Problem #272)
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中文总结 AI 辅助
研究具有算术级数交集的整数族中\(t(N)\)的值，通过穷举计算确定\(3\leq N\leq12\)时\(t(N)\)，猜想\(t(N)=\binom{N}{X}+1+\lfloor(N - 1)/Y\rfloor\)，结合计数不等式和结构定理证明相关结论并给出结构约束。
中文摘要 AI 辅助
设\(t(N)\)是最大的\(t\)，对于此存在不同的集合\(A_1,\dots,A_t\subseteq\{1,\dots,N\}\)，使得对于所有\(i\neq j\)，\(A_i\cap A_j\)是一个非空算术级数（埃尔德什问题#272）。西蒙诺维茨和绍斯证明了\(t(N)=O(N^2)\)并猜想\(\binom{N}{2}+1\)是最优的；绍博通过一个构造证明了\(t(N)\geq\binom{N}{2}+1+\lfloor(N - 1)/4\rfloor\)，证明了渐近式\(t(N)=N^2/2 + O(N^{5/3}(\log N)^3)\)，并询问\(t(N)=\binom{N}{2}+O(N)\)是否成立以及是否某些元素位于任何极值族的所有集合中（核问题）。我们通过穷举计算精确确定了所有\(3\leq N\leq12\)时的\(t(N)\)：在整个范围内绍博的下界是精确的，并且我们猜想对于每个\(N\)，\(t(N)=\binom{N}{2}+1+\lfloor(N - 1)/4\rfloor\)。为了得到匹配的上界，我们证明对于每个\(N\)，绍博的界是所有具有公共元素的族（星族）中的精确最大值。证明结合了一个关于阶梯区域的自包含的“缺陷一”计数不等式和一个新的结构定理：这样一个族的每个非级数成员都包含一个其他成员不能共享的坏对。因此，细化的猜想简化为一个剩余的陈述，即绍博的核猜想，即某些元素位于极值族的所有集合中，并且我们证明了关于假定非星极值族的第一个结构约束。
英文摘要
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.
