Four-term progression hypergraph encoding ($L_i={\sum }_{j\ne i}(j-i)x_j$)

Use four parts indexed by $0,1,2,3$ in a cyclic group $G$, and include the triple missing part $i$ when $L_i\in A$. A transversal three-uniform tetrahedron then has the four values $S_1-i{S}_0$, where $S_0={\sum }_j{x}_j$ and $S_1={\sum }_{j}j{x}_j$, so it encodes a four-term arithmetic progression. Constant progressions give $|A||G{|}^2$ edge-disjoint tetrahedra. This is the bridge from the tetrahedron removal lemma to the Szemerédi theorem.