Undecidability of polynomial inequalities in weighted graph homomorphism densities

TL;DR

This paper proves that determining the validity of polynomial inequalities in graph homomorphism densities is undecidable for a broad class of kernels, extending previous results and addressing open questions in extremal combinatorics.

Contribution

It extends the undecidability result to kernels with ranges beyond [0,1], specifically those containing all kernels with range {0,a}, and answers a question by Lovász.

Findings

Undecidability of polynomial inequalities for kernels with range {0,a}

Extension of previous undecidability results to broader kernel classes

Addresses open problem about certificates for homomorphism density inequalities

Abstract

Many problems and conjectures in extremal combinatorics concern polynomial inequalities between homomorphism densities of graphs where we allow edges to have real weights. Using the theory of graph limits, we can equivalently evaluate polynomial expressions in homomorphism densities on kernels $W$, i.e., symmetric, bounded, and measurable functions $W$ from $[0,1]^2 \to \mathbb{R}$. In 2011, Hatami and Norin proved a fundamental result that it is undecidable to determine the validity of polynomial inequalities in homomorphism densities for graphons (i.e., the case where the range of $W$ is $[0,1]$, which corresponds to unweighted graphs, or equivalently, to graphs with edge weights between $0$ and $1$). The corresponding problem for more general sets of kernels, e.g., for all kernels or for kernels with range $[-1,1]$, remains open. For any $a > 0$, we show undecidability of polynomial inequalities for any set of kernels which contains all kernels with range $\{0,a\}$. This result also answers a question raised by Lov\'asz about finding computationally effective certificates for the validity of homomorphism density inequalities in kernels.