good. I don’t recognize who you are but certainly you are going to

a famous blogger if you happen to are not already. Cheers! ]]>

http://www.ams.org/notices/201104/rtx110400550p.pdf ]]>

http://www.ams.org/publications/journals/notices/201607/rnoti-p732.pdf

]]>To obtain the promised conclusion of the lemma 7, as we can just apply a suitable bijection X->Y to map the pair (X,P’) to a pair (Y,P). (And rename Y to X in the current notation of the Lemma.)

]]>Regarding the number of points: the construction in Lemma 7 creates a convexity space with |N| points where N is the nerve. The nerves in question are HUGE though. More precisely, look at the proof of theorem 2 in Section 2. Let K=3(k-1)+1. There are K of A-families, 3*binom(K,4) B-families and binom(K,2) C-families. We then apply the closure operator (denoted by hat) to each of these families, to obtain A-hat, B-hat, C-hat families. Then N consists of these families and **their subfamilies**. For example, each C-hat-family has size 2^K-(binom(K,2)-1)-K-1=2^K-K(K+1)/2, and so there are 2^{2^K-K(K+1)/2} subfamilies of each C-hat-family. Each of them gives a point in the resulting convexity space. For k=3, we thus get 2^100 points from each C-set alone. That is a lot of points!

It is very likely that one can significantly reduce the size of these counterexamples. Back when I wrote the paper, I believed I could reduce the number of points somewhat by making the construction somewhat messier, but I saw no way to reduce the number of points to polynomial in k, say.

I hope this answers your questions. Let me know if you have more.

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