Polynomial families of incident flags and explicit off-diagonal Ramsey graphs

Abstract

We construct explicit off-diagonal Ramsey graphs from polynomial families of incident point--hyperplane flags. Universal interpolation translates the exact dimension of ordered clique configurations into a bound on their coefficient--label incidence. At the critical dimension, a polynomial separates the forbidden-pair image from its diagonal; deleting the corresponding edges preserves a triangular rank certificate. Restriction of scalars realizes the rational endpoint without rounding loss. The resulting fixed-$s$ exponent has leading scale $s/(2{\log }_{2}s)$, and short Frobenius relations give the explicit example $R(16,t)\ge \Omega (t^{2.0539221767\dots })$. A finite-fiber construction gives the range $s^2(\log s{)}^2=o(\log t)$. For fixed parameters, algebraic preprocessing terminates and field initialization, vertex decoding and adjacency take deterministic time polynomial in the extension degree. We also retain an elimination-free variant and formulate the filtering argument for general forbidden configurations with a distinguished edge.

arXiv comment27 pages, 1 table