Acyclic sets of linear orders: A progress report

Peter C. Fishburn ()
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Peter C. Fishburn: AT&T Shannon Laboratory, Florham Park, NJ 07932, USA

Social Choice and Welfare, 2002, vol. 19, issue 2, 447 pages

Abstract: Let f(n) be the maximum cardinality of an acyclic set of linear orders on {1, 2, \dots , n}. It is known that f(3)=4, f(4)=9, f(5)=20, and that all maximum-cardinality acyclic sets for n\leq 5 are constructed by an "alternating scheme". We outline a proof that this scheme is optimal for n=6, where f (6)=45. It is known for large n that f (n) >(2.17)n and that no maximum-cardinality acyclic set conforms to the alternating scheme. Ran Raz recently proved that f (n) 0 and all n. We conjecture that f (n + m)\leqf (n + 1) f (m + 1) for n , m\geq 1, which would imply f (n)

Date: 2002-04-10
Note: Received: 12 April 2000/Accepted: 4 December 2000
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