{"id":"hexagon/2610.00206","pdf":"https://hexagonmath.org/pdf/2610.00206v1","url":"https://hexagonmath.org/abs/2610.00206v1","tags":["msc:05D05","msc:05C35"],"score":0,"title":"A 23-Point Counterexample to a Modified Half-Frequency Conjecture for Union–Intersection Systems","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Jungang Chen"],"reviews":0,"subject":"math.CO","summary":"We consider a variant of Vince's half-frequency conjecture for union-intersection systems in which an entire system must be nontrivial, while a system is declared trivial only when all its blocks are singletons or when its only block is the whole ground set; every such trivial system is stipulated to be irreducible. For nontrivial systems, irreducibility retains Vince's two-part partition criterion, and complete irreducibility is tested on the system and all subsystems induced by actual blocks. We give a system on 23 points, generated by 17 three-element sets, whose union-intersection…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-10-12","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}