#196: two exact monotone 4-AP avoider counts
The target (erdosproblems.com/196, OPEN). Must every permutation of the positive integers contain four values in arithmetic progression whose positions are increasing or decreasing? The infinite question remains open.
A finite statistic. Let theta4(n) count permutations of {1,...,n} with no monotone four-term arithmetic-progression subsequence. Three complete exact enumerations now give
theta4(13)=949,812,334
theta4(14)=9,471,574,188.
Prior exact data. Goh and Zhao's peer-reviewed 2021 paper publishes an equivalent longest-progression distribution through n=10. The public arXiv v2 table also contains rows n=11 and n=12, even though nearby prose says the table stops at 10. Summing its columns for longest progression lengths 1, 2, and 3 reproduces theta4(n) through n=12. The two displayed values above are therefore the only new finite frontier claimed here, and the dated failure to find them in prior tables is search-limited, not an assertion of absolute priority.
Why the computation is exact. The first program appends unused values and rejects precisely when the new last position completes an increasing or decreasing four-term progression; a value-complement involution pairs its search subtrees. It was rebuilt and rerun through n=14, with a direct all-permutations check through n=10.
The hostile audit then reached n=14 twice without using that search tree. One implementation tracks progress in every oriented forbidden word and uses no complement symmetry. Another builds every avoider by inserting the new maximum into every possible position. Both independently return the same 9,471,574,188 count. Exhaustive Python checks through n=9, sanitizer runs, five OpenMP thread counts, and the identity n!A4(n)/12 for total forbidden-progression incidences supply additional controls.
Why this does not solve Problem 196. Davis, Entringer, Graham, and Simmons already constructed a doubly-infinite avoiding order, so finite avoiders exist for every n without these counts. The missing step is an avoiding order of type omega: in a singly infinite permutation every element has only finitely many predecessors. Finite enumeration and ordinary compactness do not provide that order-type control.
The live page showed no claimed proof or current worker on 11 August 2026. The frozen package records the original computation, the corrected literature scope, two independent n=14 enumerators, exact logs, sources, and a hostile audit. Two non-load-bearing site snapshots were excluded because one captured uncommitted ledger work unrelated to this problem.