{"id":"palomar/PALOMAR-2026-10-28-000017","url":"https://github.com/Eyali1001/scc-group-barrier/tree/6e572ed4367b37cfd89830f2fa99f4705d300c93","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-28-000017-v1.json","repo":"https://github.com/Eyali1001/scc-group-barrier","tags":["cs.CC","math.CO","math.GR","msc:68Q17","msc:15A69","msc:20D60"],"score":0,"title":"Group constructions cannot refute the Set Cover Conjecture, including tensor powers","trust":"high","domain":"cs.CC","source":"palomar","authors":["Eyal Merhavi"],"reviews":0,"subject":"cs.CC","summary":"Pratt (STOC 2024, arXiv:2311.02774, Corollary 1.12) showed that if the Set Cover Conjecture holds, then for every k the asymptotic rank of the balanced tripartitioning tensor T_k is at least tau_k = 8^k * C(3k,k) * C(2k,k) / 27^k. A natural way to bound this rank, and Pratt's own route to rank at most 8^k/2, is a group construction: maps f1, f2, f3 from k-subsets of a 3k-set to a group G with f1(A) f2(B) f3(C) = 1 exactly when A, B, C are pairwise disjoint (a partition analogue of the Cohn-Umans triple product property). We formalise that no such construction, in any finite group, abelian or…","category":"research","comments":0,"disputed":false,"division":"cs","mentions":0,"verified":true,"discussed":0,"published":"2026-10-28","reactions":{},"formalized":true}