A diameter-free Property B bound for arbitrary van der Waerden sets

Definitions and mathematical statement

For $k\ge2$, a $k$-term arithmetic progression means $\{a,a+d,\ldots,a+(k-1)d\}\subseteq\mathbb Z$ with $d>0$. A finite set $X\subseteq\mathbb Z$ is two-color $k$-progression forcing if every coloring $X\to\{\mathrm{red},\mathrm{blue}\}$ contains a monochromatic progression of this form entirely within $X$. Let \[ W^*(k)=\min\{|X|:X\subseteq\mathbb Z\text{ is finite and two-color }k\text{-progression forcing}\}. \] Also let $W(k)$ be the minimum $N$ for which $[N]$ is two-color $k$-progression forcing. These numbers are finite, and $W^*(k)\le W(k)$ because intervals are among the permitted sets. The extremal set for $W^*(k)$ need not be an interval and has no prescribed diameter or maximum element.

The two questions are whether the nonnegative difference $W(k)-W^*(k)$ is unbounded as $k\to\infty$, and whether \[ \lim_{k\to\infty}\frac{W^*(k)}{W(k)}=1. \] Unbounded difference means that no fixed constant bounds the gaps for all $k$; it does not assert that the gap increases monotonically or tends to infinity along every sequence. The two proposed behaviours can hold together, since an unbounded additive saving may still be a vanishing relative saving. Both questions compare the smallest cardinality of any forcing set with the forcing threshold for intervals.

Short English statement

How much can arbitrary integer sets save over intervals in forcing monochromatic progressions: is the additive saving unbounded, and does the relative saving vanish?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001090 (AIM-COMBINATORICS-0215), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Recent trends in additive combinatorics, item 1.9. Original workshop question underlying AIM-COMBINATORICS-0215. https://aimath.org/WWN/additivecomb/additivecomb.pdf.