In the last post I concentrated on examples, so in this one I’ll concentrate on conjectures related to FUNC, though I may say a little about examples at the end, since a discussion has recently started about how we might go about trying to find a counterexample to FUNC.
A proposal for a rather complicated averaging argument
After the failure of the average-overlap-density conjecture, I came up with a more refined conjecture along similar lines that has one or two nice properties and has not yet been shown to be false.
The basic aim is the same: to take a union-closed family and use it to construct a probability measure on the ground set in such a way that the average abundance with respect to that measure is at least 1/2. With the failed conjecture the method was very basic: pick a random non-empty set and then a random element .
The trouble with picking random elements is that it gives rise to a distribution that does not behave well when you duplicate elements. (What you would want is that the probability is shared out amongst the duplicates, but in actual fact if you duplicate an element lots of times it gives an advantage to the set of duplicates that the original element did not have.) This is not just an aesthetic concern: it was at the heart of the downfall of the conjecture. What one really wants, and this is a point that Tobias Fritz has been emphasizing, is to avoid talking about the ground set altogether, something one can do by formulating the conjecture in terms of lattices, though I’m not sure what I’m about to describe does make sense for lattices.
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