A collection F of subsets of \ 1,2, ⋯ ,n\ is a dual intersecting system if no two sets in F have an empty intersection or an exhaustive union. Let fⱼ be the number of sets in F that contain point j, and let fⱼL be the number of sets not in F but included in some set in F that contain j. We conjecture that if F is a dual intersecting system, then fⱼL fⱼ for some j in \ 1, ⋯ ,n\. This is shown to be true if either min fⱼ n or min fⱼ < 8.
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P. C. Fishburn (1987) studied this question.
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