# Simple Graph Density Inequalities with no Sum of Squares Proofs

## TL;DR

This paper identifies conditions where certain graph density inequalities cannot be proved using sum of squares methods, providing new insights into the limitations of this certification approach in extremal combinatorics.

## Contribution

The paper introduces a simple criterion for non-representability of graph density inequalities as sums of squares and applies it to key conjectures and inequalities in the field.

## Key findings

- Blakley-Roy inequality lacks a sum of squares certificate for odd path lengths.
- Certain inequalities cannot be certified by sums of squares with specific multipliers.
- The results extend to frameworks by Razborov and Lovász-Szegedy.

## Abstract

Establishing inequalities among graph densities is a central pursuit in extremal combinatorics. A standard tool to certify the nonnegativity of a graph density expression is to write it as a sum of squares. In this paper, we identify a simple condition under which a graph density expression cannot be a sum of squares. Using this result, we prove that the Blakley-Roy inequality does not have a sum of squares certificate when the path length is odd. We also show that the same Blakley-Roy inequalities cannot be certified by sums of squares using a multiplier of the form one plus a sum of squares. These results answer two questions raised by Lov\'asz. Our main tool is used again to show that the smallest open case of Sidorenko's conjectured inequality cannot be certified by a sum of squares. Finally, we show that our setup is equivalent to existing frameworks by Razborov and Lov\'asz-Szegedy, and thus our results hold in these settings too.