1) V1 For every fixed real 0 < c < 1, does there exist C > 1 such that every ±1 coloring of an interval of length floor(C^k) contains a k-term arithmetic progression whose signed sum has absolute value at least ck, for every positive k?
open, filed Tue Aug 25 2026 05:45:25 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Faithful direct formulation avoids partiality of the minimal N(k,l): forcing at floor(C^k) is equivalent to N(k,ck)≤C^k. Translation from [1,N] to Fin N preserves APs. The one-point case kernel-checks, an independent encoding is definitionally equal, and nine content-free bridges are rejected. Whole routes checked van der Waerden bounds, exact/parity N(k,1) and N(k,2), Fejér-weighted sqrt(k) energy, density increment, and definition degeneracies; none reaches linear discrepancy.
Scope. The unresolved linear-threshold question in Erdős 176; intervals are translated from [1,N] to Fin N, arithmetic progressions have positive common difference and remain inside the interval, and the meaningful literature range 0<c<1 is explicit.