Monotone 4-term Arithmetic Progressions

Definitions and mathematical statement

Let $\mathbb N=\{1,2,\ldots\}$ and let $\pi:\mathbb N\to\mathbb N$ be a bijection. For positive integers $a,d$, the four indices $a,a+d,a+2d,a+3d$ form a nonconstant arithmetic progression. Call this progression monotone for $\pi$ if either \[ \pi(a)<\pi(a+d)<\pi(a+2d)<\pi(a+3d) \] or all three inequalities are reversed. All inequalities are strict, as the values of a permutation are distinct.

The question is whether every permutation $\pi$ admits such $a,d$. Equivalently, regard a permutation as a one-sided sequence $b_1,b_2,\ldots$ listing every positive integer exactly once. Must four terms occurring in their sequence order form an increasing or decreasing arithmetic progression? To relate these formulations, let $\pi(t)$ be the position at which the integer $t$ occurs; the inverse of any permutation is again a permutation.

The positions of the four terms in the sequence formulation need not be equally spaced. In the displayed formulation the indices are equally spaced and the values are merely monotone, not necessarily an arithmetic progression. This distinction prevents imposing an extra requirement absent from the problem. The ordering has order type $\mathbb N$, so arrangements indexed by all integers do not automatically answer it. A counterexample must be a single bijection avoiding every increasing and every decreasing four-term arithmetic progression in this sense.

Short English statement

Does every one-sided ordering of the positive integers contain four arithmetic-progression values appearing in increasing or decreasing order?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3040 (OPG-618), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Monotone 4-term Arithmetic Progressions. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/monotone_4_term_arithmetic_progressions.