An upper bound on tricolored ordered sum-free sets

2017/08/24 by Taegyun Kim, Kim, Taegyun, Sang‐il Oum +1
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #Complexity and Algorithms in Graphs #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1708.07263

openalex publication_date 2017/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present a strengthening of the lemma on the lower bound of the slice rank by Tao (2016) motivated by the Croot-Lev-Pach-Ellenberg-Gijswijt bound on cap sets (2017, 2017). The Croot-Lev-Pach-Ellenberg-Gijswijt method and the lemma of Tao are based on the fact that the rank of a diagonal matrix is equal to the number of non-zero diagonal entries. Our lemma is based on the rank of upper-triangular matrices. This stronger lemma allows us to prove the following extension of the Ellenberg-Gijswijt result (2017). A tricolored ordered sum-free set in \mathbb F_pⁿ is a collection \(a_i,b_i,c_i):i=1,2,…,m\ of ordered triples in (\mathbb F_pⁿ )³ such that a_i+b_i+c_i=0 and if a_i+b_j+c_k=0, then i≤ j≤ k. By using the new lemma, we present an upper bound on the size of a tricolored ordered sum-free set in \mathbb F_pⁿ.

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