Common and Sidorenko equations in Abelian groups

DOI10.4310/JOC.2023.V14.N1.A3arXiv2109.04445MaRDI QIDQ6377260

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Publication date: 9 September 2021

Abstract: A linear configuration is said to be common in a finite Abelian group

G

if for every 2-coloring of

G

the number of monochromatic instances of the configuration is at least as large as for a randomly chosen coloring. Saad and Wolf conjectured that if a configuration is defined as the solution set of a single homogeneous equation over

G

, then it is common in

m a t h b b F_{p}^{n}

if and only if the equation's coefficients can be partitioned into pairs that sum to zero mod

p

. This was proven by Fox, Pham and Zhao for sufficiently large

n

. We generalize their result to all sufficiently large Abelian groups

G

for which the equation's coefficients are coprime to

v e r t G v e r t

Mathematics Subject Classification ID

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