Undecidability of linear inequalities in graph homomorphism densities

Authors:
Hamed Hatami and Serguei Norine

Journal:
J. Amer. Math. Soc. **24** (2011), 547-565

MSC (2010):
Primary 05C25, 05C35, 12L05

Published electronically:
December 3, 2010

MathSciNet review:
2748400

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Abstract: The purpose of this article is to show that even the most elementary problems in asymptotic extremal graph theory can be highly non-trivial. We study linear inequalities between graph homomorphism densities. In the language of quantum graphs the validity of such an inequality is equivalent to the positivity of a corresponding quantum graph. Similar to the setting of polynomials, a quantum graph that can be represented as a sum of squares of labeled quantum graphs is necessarily positive. Lovász (Problem 17 in his manuscript *Graph homomorphisms: Open problems*) asks whether the opposite is also true. We answer this question and also a related question of Razborov in the negative by introducing explicit valid inequalities that do not satisfy the required conditions. Our solution to these problems is based on a reduction from real multivariate polynomials and uses the fact that there are positive polynomials that cannot be expressed as sums of squares of polynomials.

It is known that the problem of determining whether a multivariate polynomial is positive is decidable. Hence it is very natural to ask ``Is the problem of determining the validity of a linear inequality between homomorphism densities decidable?'' We give a negative answer to this question which shows that such inequalities are inherently difficult in their full generality. Furthermore we deduce from this fact that the analogue of Artin's solution to Hilbert's seventeenth problem does not hold in the setting of quantum graphs.

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Additional Information

**Hamed Hatami**

Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Address at time of publication:
School of Computer Science, McGill University, Montréal, Quebec, Canada

Email:
hatami@cs.mcgill.ca

**Serguei Norine**

Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Email:
snorin@math.princeton.edu

DOI:
http://dx.doi.org/10.1090/S0894-0347-2010-00687-X

Keywords:
Graph homomorphism density,
quantum graph,
decidability,
Artin’s theorem

Received by editor(s):
June 20, 2010

Received by editor(s) in revised form:
August 29, 2010

Published electronically:
December 3, 2010

Additional Notes:
The first author was supported in part by NSERC

The second author was supported in part by NSF under Grant No. DMS-0701033

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.