Some common names which are used in literature for such families: partial linear spaces, near linear spaces, linear hypergraphs.

This is the same result as the Fisher's inequality.

Not sure about the bound in general but there are some results about existence of sub-planes that are known. For example, combinatorics.org/ojs/index.php/eljc/article/view/v18i1p2. For details on small planes you can also refer to uwyo.edu/moorhouse/pub/planes

I have found a set of 200 pairwise disjoint spreads in McL. So $s_5(McL) \geq 200$. There might be a full resolution but I stopped my search at 200 as it seemed like it was taking too long. I can send you the SAGE code for this if you want. I used a rather brute force way of doing it using the inbuilt implementation of dancing links algorithm in SAGE.

@FelixGoldberg: It seems quite interesting to me. Have you tried to compute the five different spreads they mention in the paper? If you want then you can send more details about what you are working on and trying to achieve via email, [email protected].

Why 7? Do you have an example of a partial 5-resolution of size 7 for McL?

Are you aware of this lower bound? homepages.cwi.nl/~lex/files/countpms2.pdf

There is an improvement on the result mentioned in the paper of A. Chowdhury which is mentioned here: arxiv.org/abs/1405.0909

Thanks for correcting me. It should be that the number of classes of central involutions is at most the number of involutions in the center of a Sylow 2-subgroup. Since given a central involution f, if H is a Sylow 2-subgroup contained in the normalizer of f then f must itself belong to H as otherwise, H and f would generate a subgroup of order 2|H|.

But I suppose those groups are too special and this result won't be true in general.