If is the maximal size of a tower, is $\latex \mu$ also the maximal size of a family such that has no pseudo-intersection, and such that is injective. (where denote the set of all f-lower bound ("pseudo intersections") of a family )

Such an necessary has the fip. Note that we can obtain such a family from any fip family without pseudo intersection by considering the family of the members of that satisfy

if this could be "kind of" an indication to the way to build a tower from any with fip and npi.

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For example by choosing $ latex C_i\in i_f(\left\{F_i,\, i<j\right\}$ for all and compose it with th operation that is defined here https://gowers.wordpress.com/2017/09/19/two-infinities-that-are-surprisingly-equal/#comment-203525

The main difference is instead of composing the $F_i$ we compose the that belong to members of a decreasing family of co-filters* with empty (usual) intersection., so that we can hope in a more realistic way that the tower we get has no pseudo -intersction (npi)

I will develop in a MSE post some ideas I've got about the way to hope to make it work or at least exhibe a reason why it's not easy by reducing the aim to special cases to discuss : one idea could be to use the preorder that is called in the paragraph "motivation" of this link on MSE : https://math.stackexchange.com/questions/2577440/decreasing-family-of-families-with-lower-bound-zero (I already wrote this in upper posts but note that $ latex A*B<_b A$ and also )

(*the set of complements of is a filter for any family $X$, so I qualify a co-filter, but I don't know if it is the right terminology)

]]>https://mathoverflow.net/q/41214/93609

I gave quite a detailed answer to it (surprising myself a little in doing so), which shows that I have started putting into practice some of my thoughts about GitHub issues, etc, outlined in my reply above:

https://mathoverflow.net/a/288852/93609

There is still a way to go, years probably, but perhaps things are far enough advanced, or soon will be, for me to mention it (somewhat sheepishly) in a place where it might get some attention.

So many attempts at this kind of thing seem to peter out (again see the answers to the mathoverflow,com question in the unlikely event that you need any convincing). I guess my excuse for mentioning my own attempt now, therefore, at the risk of appearing to be too self-aggrandising, is that I don’t intend to let that happen.

]]>Also such sets have been designed that “for any three players, there is a fourth that beats all of them”. Yet “the optimal graph is unknown for the 5 player game and above” http://www.mathpuzzle.com/MAA/39-Tournament%20Dice/mathgames_07_11_05.html

Interestingly–can increasing complexity of dice construction for N player games be related to The P versus NP problem https://en.wikipedia.org/wiki/P_versus_NP_problem?

Can answering this question be aimed for further results?

” Can the choice of be made totally arbitrarily in the construction? If I choose for every in the maximally spanning set, we would have constructed a linear map.”

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