AI 中文总结
本文针对Erdős-Graham两集合置换问题，通过研究二元候选集S_A的结构限制，证明其不存在可允许置换，表明典型二元划分无法解决该问题，该问题仍待解决。
AI 中文摘要
正整数集合的置换若在位置顺序中不包含递增或递减的三项等差数列，则称为可允许置换。Davis、Entringer、Graham和Simmons证明了正整数无法进行可允许置换，将其划分为三个可允许置换集合，并提出是否只需两个集合的问题。本文研究典型的二元候选集S_A，即所有偶数k≥2对应的区间(2^{k-1},2^k]的并集。我们建立了可允许置换的若干结构限制，包括轨道障碍、非对称平衡定律，以及无法将所有足够大的二元块作为连续序列放置。对于主要结果，我们将S_A的可允许性简化为区间(M,2M]上的顺序小工具的可行性：该顺序必须避免单调三项等差数列，并满足由15和16的值诱导的15个优先约束。我们利用等差数列梯子上的锯齿传播、转移锁和镜像洪水归纳法，证明该小工具的三约束核心对每个模8余0的M均不一致。由于所有相关二元尺度均属于该剩余类，S_A不存在可允许置换。因此，典型二元划分未能解决两集合问题，该问题仍未解决。独立的SAT编码和证书检查为人类可读的证明提供了辅助验证。
英文摘要
Call a permutation of a set of positive integers admissible if it contains no increasing or decreasing three-term arithmetic progression in position order. Davis, Entringer, Graham, and Simmons proved that the positive integers are not admissibly permutable, partitioned them into three admissibly permutable sets, and asked whether two sets suffice. We study the canonical dyadic candidate S_A, the union of the blocks (2^{k-1}, 2^k] over even k >= 2. We establish several structural restrictions on admissible permutations, including an orbit obstruction, an asymmetric balance law, and the impossibility of placing all sufficiently large dyadic blocks as contiguous runs. For the main result, we reduce admissibility of S_A to feasibility of an order gadget on (M, 2M]: the order must avoid monotone three-term progressions and satisfy fifteen precedence constraints induced by the values 15 and 16. We then prove that a three-constraint core of this gadget is inconsistent for every M congruent to 0 mod 8, using zigzag propagation on arithmetic-progression ladders, a transfer lock, and mirror-flood induction. Since every relevant dyadic scale lies in this residue class, S_A admits no admissible permutation. Thus the canonical dyadic partition does not solve the two-set problem, which remains open. Independent SAT encodings and certificate checks provide auxiliary verification of the human-readable proof.
Comments21 pages. Verification scripts, solver logs, DRAT certificates, and a one-command reproduction kit at https://github.com/Wkasel/erdos197 (immutable tag arxiv-v1). Prepared with machine assistance, disclosed in the paper
