An upper bound on tricolored ordered sum-free sets

Understand

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).