I saw yesterday that this crossed in the mail with the published
solution, which stated that P(15)=0 because 14c2>81, though you noted
that according to the proposer, P(20)>0.
The fallacy in the published solution is that the solver assumed that
each base pair corresponded to a unique card that would complete a
Set. However, if you have four or more cards, it's possible for two
or more of the base pairs to be completed by the same card. If you
choose four cards chosen at random, usually their connections will map
as a tetrahedron, with each edge representing one of the six base
pairs, and their respective midpoints representing the cards
completing six different Sets. However, given any three cards that
don't form a Set, there will be three other cards in the deck with the
property that if you add that card to the original three, the
connections will map as the corners of a square, with the diagonals
crossing at the center. This makes the published probabilities too
low for all N>4. (Together with the three original cards, and the
cards that complete Sets for the three original base pairs, these
three cards will complete a copy of the 9-card deck for the simplified
two-attribute version of the game, though the two new "attributes" are
likely to be a complex combination of the original four.)
As I had noted in my original incomplete attempt at a solution, it's
easy to describe a 16-card configuration with no Set. I described it
geometrically at the time, but in terms of the actual cards, you can
construct it by throwing out one value for each attribute - for
example, eliminate the 65 cards that have either the number 2, the
striped shading, the squiggle figure, or the color purple. Within
this configuration, 8 different base pairs will cross and be completed
by the same one card at the center - in this case, the card with two
striped purple squiggles.
That 16-card configuration is complete, meaning no additional cards
can be added without making a Set, but it's not maximal. At the time,
the best configuration I had found with no Set had 18 cards, but I was
fairly confident one could do better. After I sent it, I found the
20-card configuration below (with 10 base pairs intersecting at the
center). I was hoping to prove that it was optimal (or find a
counterexample), but eventually gave up and sent you what I had. I'm
guessing it's the same configuration the proposer found, but I don't
know whether the proposer was able to prove that it's the unique
20-card configuration (subject to the arbitrary choice of center and
axes).
Finding all possible 20-card configurations (or even asking whether
the proposer was correct that such a configuration exists, since there
don't seem to be a lot of readers who have gotten even that far yet)
could be offered as a follow-up puzzle.
If my guess that this configuration is unique is correct, then because
it's obviously complete, it must also be maximal, which would mean
P(21) would be 0, and if I've counted the symmetries correctly, then
P(20) would be 81*80*78*72*54 / (4! * 81c20), which if I've done the
arithmetic correctly is 972 / 79*37*73*71*23*67*11*31 ≈ .00000001221%