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 Lovász. 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 Lovász-Szegedy, and thus our results hold in these settings too.
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Grigoriy Blekherman was partially supported by NSF grant DMS-1352073. This material is partially based upon work supported by the National Science Foundation under Grant No. 1440140, while Annie Raymond and Rekha Thomas were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the fall of 2017. Mohit Singh was partially supported by NSF grant CCF-1717947 and Rekha Thomas was partially supported by NSF grant DMS-1719538.
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Blekherman, G., Raymond, A., Singh, M. et al. Simple Graph Density Inequalities with No Sum of Squares Proofs. Combinatorica 40, 455–471 (2020). https://doi.org/10.1007/s00493-019-4124-y
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