Abstract

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, that is, 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, for example, 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 Lovász about finding computationally effective certificates for the validity of homomorphism density inequalities in kernels.

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.