= Four-term progression hypergraph encoding
{title2=$L_i=\sum_{j\ne i}(j-i)x_j$}
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 $S_1-iS_0$, where $S_0=\sum_jx_j$ and $S_1=\sum_jjx_j$, 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>.
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