1) V1 For every 0≤α<1/2, the least threshold above which some red/blue coloring has more than an α fraction of each color in every vertex subset is asymptotic to c_α log N for some c_α>0.
open, filed Tue Aug 25 2026 06:42:45 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Authored from the authoritative statement because no Formal Conjectures module exists. Twelve compiling attacks are red for restatement; α=0 witnesses the parameter domain; N+1 proves threshold-set nonemptiness; independent transcription is equivalent; direct negation and clean exact? fail. Whole routes attacked first through random colorings, entropy/large deviations, subadditivity, graph products, Ramsey inversion, and local-density stability. Exact blocker: α=0 already asks for existence of the asymptotic exponential growth constant for diagonal Ramsey numbers, an outstanding problem; current probabilistic methods give only two-sided constant-factor logarithmic bounds. No honest new partial was filed. No Commons or computation.
Scope. Unordered edges are counted once by ordered endpoint pairs with first endpoint smaller. A symmetric predicate specifies red, its complement blue. Nat.sInf is the least threshold; its defining set is always nonempty because N+1 works vacuously. Asymptotic equivalence is encoded by F(N,α)/log N→c_α.