Simultaneously avoiding bounded weighted averages

Definitions and mathematical statement

Fix positive integers $t$ and $N$, and write $[N]=\{1,\ldots,N\}$. Consider a subset $A\subseteq[N]$ with the following simultaneous avoidance property: for every pair of integers $1\le m,n\le t$, the equation \[ mx+ny=(m+n)z \] has no nontrivial solution with $x,y,z\in A$. Here a solution is trivial exactly when $x=y=z$. Since $m,n$ are positive, any solution with two equal variables is already trivial; a prohibited solution therefore uses three distinct elements. The coefficients are ordinary integers, and equality is in $\mathbb Z$, without reduction modulo any number.

Define \[ F_t(N)=\max\{|A|:A\subseteq[N]\text{ satisfies all the above avoidance conditions}\}. \] The problem is to determine how large $F_t(N)$ can be, particularly its asymptotic growth as $N\to\infty$ with $t$ fixed and the dependence of useful estimates on $t$. The same set $A$ must avoid all $t^2$ equations at once; choosing a different set for each coefficient pair is not sufficient. For $m=n=1$, the excluded configuration is an ordinary three-term arithmetic progression, while other coefficient pairs forbid further rational weighted averages between two selected integers. Pairs of proportional coefficients can specify the same equation, but retaining them does not affect the definition. Both the ambient interval and the cardinality objective are part of the extremal question.

Short English statement

How large can an interval subset be if it simultaneously avoids every nontrivial three-variable weighted-average equation with coefficients at most t?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001112 (AIM-COMBINATORICS-0237), 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 2.12. Original workshop question underlying AIM-COMBINATORICS-0237. https://aimath.org/WWN/additivecomb/additivecomb.pdf.