Simple Graph Density Inequalities with No Sum of Squares Proofs


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.

This is a preview of subscription content, log in to check access.


  1. [1]

    G. R. Blakley and P. Roy: A Hölder type inequality for symmetric matrices with nonnegative entries, Proceedings of the American Mathematical Society16 (1965), 1244–1245.

    MathSciNet  MATH  Google Scholar 

  2. [2]

    G. Blekherman, P. Parrilo and R. R. Thomas: Semidefinite Optimization and Convex Algebraic Geometry, SIAM, 2012.

    Google Scholar 

  3. [3]

    G. Blekherman, A. Raymond, M. Singh and R. R. Thomas: Homogeneous graph density inequalities and sums of squares, in preparation.

  4. [4]

    A. Carbery: A multilinear generalisation of the Cauchy-Schwarz inequality, Proceedings of the American Mathematical Society132 (2004), 3141–3152.

    MathSciNet  Article  Google Scholar 

  5. [5]

    D. Conlon and J. Lee: Sidorenko’s conjecture for blow-ups, arXiv preprint arXiv:1809.01259, 2018.

    Google Scholar 

  6. [6]

    V. Falgas-Ravry and E. R. Vaughan: Applications of the semi-definite method to the Turán density problem for 3-graphs, Combin. Probab. Comput.22 (2013), 21–54.

    MathSciNet  Article  Google Scholar 

  7. [7]

    T. Gowers and J. Long: Entropy and Sidorenko’s conjecture after Szegedy,

  8. [8]

    J. Han Kim, C. Lee and J. Lee: Two approaches to Sidorenko’s conjecture, Transactions of the American Mathematical Society368 (2016), 5057–5074.

    MathSciNet  Article  Google Scholar 

  9. [9]

    H. Hatami, J. Hladký, D. Král, S. Norine and A. Razborov: On the number of pentagons in triangle-free graphs, J. Combin. Theory Ser. A120 (2013), 722–732.

    MathSciNet  Article  Google Scholar 

  10. [10]

    H. Hatami and S. Norine: Undecidability of linear inequalities in graph homomorphism densities, J. Amer. Math. Soc.24 (2011), 547–565.

    MathSciNet  Article  Google Scholar 

  11. [11]

    N. H. Katz and T. Tao: Bounds on arithmetic projections, and applications to the Kakeya conjecture, Math. Res. Lett.6 (1999), 625–630.

    MathSciNet  Article  Google Scholar 

  12. [12]

    J. L. Xiang Li and B. Szegedy: On the logarithimic calculus and Sidorenko’s conjecture, arXiv:1107.1153, 2011.

    Google Scholar 

  13. [13]

    L. Lovász: Graph homomorphisms: Open problems, manuscript available at, 2008.

    Google Scholar 

  14. [14]

    L. Lovász: Large Networks and Graph Limits, volume 60 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 2012.

    Google Scholar 

  15. [15]

    L. Lovász and B. Szegedy: Limits of dense graph sequences, Journal of Combinatorial Theory, Series B96 (2006), 933–957.

    MathSciNet  Article  Google Scholar 

  16. [16]

    L. Lovász and B. Szegedy: Random graphons and a weak Positivstellensatz for graphs, J. Graph Theory70 (2012), 214–225.

    MathSciNet  Article  Google Scholar 

  17. [17]

    A. Raymond, J. Saunderson, M. Singh and R. R. Thomas: Symmetric sums of squares over k-subset hypercubes, Mathematical Programming167 (2018), 315–354.

    MathSciNet  Article  Google Scholar 

  18. [18]

    A. Raymond, M. Singh and R. R. Thomas: Symmetry in Turán sums of squares polynomials from ag algebras, Algebraic Combinatorics1 (2018), 249–274.

    MathSciNet  Article  Google Scholar 

  19. [19]

    A. A. Razborov: Flag algebras, The Journal of Symbolic Logic72 (2007), 1239–1282.

    MathSciNet  Article  Google Scholar 

  20. [20]

    A. A. Razborov: On 3-hypergraphs with forbidden 4-vertex configurations, SIAM J. Discrete Math.24 (2010), 946–963.

    MathSciNet  Article  Google Scholar 

  21. [21]

    A. F. Sidorenko: A correlation inequality for bipartite graphs, Graphs and Combinatorics9 (1993), 201–204.

    MathSciNet  Article  Google Scholar 

  22. [22]

    B. Szegedy: An information theoretic approach to Sidorenko’s conjecture, arXiv:1406.6738.

Download references

Author information



Corresponding author

Correspondence to Annie Raymond.

Additional information

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Blekherman, G., Raymond, A., Singh, M. et al. Simple Graph Density Inequalities with No Sum of Squares Proofs. Combinatorica 40, 455–471 (2020).

Download citation

Mathematics Subject Classification (2010)

  • 05C35
  • 90C22
  • 90C35