A 23-Point Counterexample to a Modified Half-Frequency Conjecture for Union–Intersection Systems
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…
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