Undecidability of Linear Inequalities in Graph Homomorphism Densities

Many fundamental theorems in extremal graph theory can be expressed as algebraic inequalities between subgraph densities. As is explained below, for dense graphs, it is possible to replace subgraph densities with homomorphism densities. An easy observation shows that one can convert any algebraic inequality between homomorphism densities to a linear inequality. Inspired by the work of Freedman, Lovász and Schrijver [FLS07], in recent years a new line of research in the direction of treating and understanding these inequalities in a unified way has emerged. Razborov [Raz07] observed that a typical proof of an inequality in extremal graph theory between homomorphism densities of some fixed graphs involves only homomorphism densities of finitely many graphs. He states in [Raz08a] that in his opinion the most interesting general open question about asymptotic extremal combinatorics is whether every true linear inequality between homomorphism densities can be proved using a finite amount of manipulation with homomorphism densities of finitely many graphs. Although this question itself is not well defined, a natural precise refinement is whether the problem of determining the validity of a linear inequality between homomorphism densities is decidable. We show that it is not. Our result in particular answers various related questions by Razborov [Raz07], Lovász [Lov08], and Lovász and Szegedy [LS09]. An interesting recent result in extremal graph theory, proved in several different forms [FLS07, Raz07, LS09], says that every linear inequality between homomorphism densities follows from the positive semi-definiteness of a certain infinite matrix. As an immediate consequence, every algebraic inequality between the homomorphism densities follows from an infinite number of certain applications of the Cauchy-Schwarz inequality. This is consistent with the fact that many results in extremal graph theory are proved by one or more tricky applications of the CauchySchwarz inequality. Lovász [Lov08] composed a collection of open problems in this area, and in Problem 17 he asks whether it is true or not that every algebraic inequality between homomorphism densities follows from a finite number of applications of this inequality. It is possible to rephrase this question in the language of quantum graphs defined by Freedman, Lovász and Schrijver [FLS07]. The validity