# ALGEBRAIC COMBINATORICS

Symmetry in Turán sums of squares polynomials from flag algebras
Algebraic Combinatorics, Volume 1 (2018) no. 2, pp. 249-274.

Turán problems in extremal combinatorics ask to find asymptotic bounds on the edge densities of graphs and hypergraphs that avoid specified subgraphs. The theory of flag algebras proposed by Razborov provides powerful methods based on semidefinite programming to find sums of squares that establish edge density inequalities in Turán problems. Working with polynomial analogs of the flag algebra entities, we prove that such sums of squares created by flag algebras can be retrieved from a restricted version of the symmetry-adapted semidefinite program proposed by Gatermann and Parrilo. This involves using the representation theory of the symmetric group for finding succinct sums of squares expressions for invariant polynomials. The connection reveals several combinatorial and structural properties of flag algebra sums of squares, and offers new tools for Turán and other related problems.

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DOI: https://doi.org/10.5802/alco.5
Classification: 05D99,  12D15,  90C22
Keywords: sums of squares, semidefinite programming, flag algebras, extremal combinatorics, symmetry
@article{ALCO_2018__1_2_249_0,
author = {Raymond, Annie and Singh, Mohit and Thomas, Rekha R.},
title = {Symmetry in {Tur\'an} sums of squares polynomials from flag algebras},
journal = {Algebraic Combinatorics},
pages = {249--274},
publisher = {MathOA foundation},
volume = {1},
number = {2},
year = {2018},
doi = {10.5802/alco.5},
zbl = {06882341},
mrnumber = {MR3856524},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.5/}
}
Raymond, Annie; Singh, Mohit; Thomas, Rekha R. Symmetry in Turán sums of squares polynomials from flag algebras. Algebraic Combinatorics, Volume 1 (2018) no. 2, pp. 249-274. doi : 10.5802/alco.5. https://alco.centre-mersenne.org/articles/10.5802/alco.5/

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