Uniformity of extremal graph-codes
Noé de Rancourt, Pandelis Dodos, Konstantinos Tyros
TL;DR
The paper investigates how extremal graph-codes exhibit pseudorandom, Fourier-uniform behavior and connects this to the density polynomial Hales–Jewett conjecture. It develops a Fourier-analytic framework based on Gowers uniformity norms and introduces central embeddings and HJ-subspaces to isolate structural components in both loopless and looped graph models. The main contributions show that extremal loopless codes with even edge counts are $U_2$-uniform, while odd-edge cases exhibit parity obstructions; additionally, extremal self-looped codes ($\mathcal{C}^\circ$-$\mathrm{HJ}$-codes) are $U_d$-uniform for all $d\ge 2$, with a quantitative route via the Tao–Ziegler inverse theorem. Together with partitioning results for nonclassical (and integer) polynomials and Ramsey-type canonization, these findings advance the understanding of density-polynomial Ramsey problems and provide analytic tools toward resolving first unknown cases of the density Hales–Jewett conjecture through extremal-code pseudorandomness.
Abstract
It is an important fact that extremal discrete structures -- that is, discrete structures of maximal size among those that avoid certain configurations -- exhibit strong pseudorandom behavior. We present instances of this phenomenon in the context of graph-codes, a notion put forth recently by Alon, as well as on related problems related to density polynomial Hales--Jewett conjecture.