Uniformity of extremal graph-codes

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.

💡 Research Summary

The paper investigates the pseudorandom properties of extremal graph‑codes and HJ‑codes, showing that extremality forces strong uniformity in the sense of Gowers’ U‑norms. A graph‑code (H‑code) is a family G of loopless graphs on a common vertex set such that for any two distinct members G₁, G₂ the symmetric difference G₁⊕G₂ is not isomorphic to any forbidden graph in a prescribed collection H. An HJ‑code adds the extra condition that the symmetric difference is only required when one graph contains the other. These notions, introduced by Alon, generalize classical extremal problems and are closely linked to the density polynomial Hales–Jewett (DPHJ) conjecture.

The authors first set up the algebraic framework: graphs are identified with indicator functions in the vector spaces F(