I'm actually a bit surprized at the answer to question 1 after provisional study:

I figured that was fairly obviously AcAd(h)4s5s, but it's not!!

That hand gives the following numbers against 7s3s5c4c, which I consider a pretty obvious second best (both runner-runners plus any 6 and not taking away any of your own outs):

Tie: 38.05%
Win: 32.93%
Loss: 29.02%

What actually seems to do the very best against the second best hand is trip aces with nut straight, runner-runner clubs (!!) and one exclusionary spade. AcAh4c5s (the heart is irrelevant and one can reverse the suits of the 4 and 5, but one needs to be a spade and one a club) vs. 7s5c4s3s gives the following percentages, which are clearly better:

Tie: 35.85%
Win: 35.61%
Loss: 28.54%

I don't see how is the second best hand

pokenum -o 7s 3s 5c 4c - 4s 5s ac ad -- as 2s 3c
Omaha Hi: 820 enumerated boards containing As 2s 3c
cards win %win lose %lose tie %tie EV
7s 3s 5c 4c 238 29.02 270 32.93 312 38.05 0.480
5s 4s Ac Ad 270 32.93 238 29.02 312 38.05 0.520

pokenum -o 7s 6s 5c 4c - 4s 5s ac ad -- as 2s 3c
Omaha Hi: 820 enumerated boards containing As 2s 3c
cards win %win lose %lose tie %tie EV
7s 6s 5c 4c 294 35.85 302 36.83 224 27.32 0.495
5s 4s Ac Ad 302 36.83 294 35.85 224 27.32 0.505

pokenum -o 3s 6s 5c 4c - ac ah 4s 5s -- as 2s 3c
Omaha Hi: 820 enumerated boards containing As 2s 3c
cards win %win lose %lose tie %tie EV
6s 3s 5c 4c 314 38.29 270 32.93 236 28.78 0.527
5s 4s Ac Ah 270 32.93 314 38.29 236 28.78 0.473

After thinking about this a little more I have to ask, how many people in the pot? That will really change things.

I'm assuming just two.

I had noticed that 4c5c6s7s does better than 7s3s4c5c, but your 6s3s5c4c really surprizes me. I'll have to think about that one. It's impossible to have against AcAd4c5s, which I'm currently claiming to be the best. But 6c3s5c4s should amount to the same thing since the straight flush is impossible in either case.

I just don't see why it does so well. I would think the ds high wrap would be better ???

I was afraid someone might ask that...

I don't really have one but was more hoping that exploring how various strong hands hold up might allow us to get there.

One might define "best" the following way:

1) A hand that is ALWAYS favorite regardless of what your opponent has

2) If there is more than one hand fitting criterion 1, then the hand that is the biggest favorite over its closest rival.

You're right that on criterion 2, it would probably be better to have a hand that has the biggest equity over the entire range of possible 54 hands on this board, but that gets really tedious to carry out.

My intention, really, was simply, first, to explore just which redraws are the very strongest and, secondly, to figure out where the boundaries lie on redraws where you still need to call.

I think it's pretty clear that against a tight opponent who raises your nuts, you have to have SOME redraw. But the question that Rhound's hand brought up for me is: Just how much redraw to you need to have? By making it HU here, I wanted to simplify it a bit but still get some idea of the "hierarchy" among various kinds of redraws.

I'll have to double-check your data (which I'm sure is correct since everything else you had corresponds exactly to what I got on the cardplayer calculator), but it's looking like 3456ds is the very best one...

And I don't really see why. True, the 3 takes away boating chances for the AA, but so does 3457. Ok, I think I may see where it's coming from: On 3457, one of the straight improvements also is a flush improvement. So there are only 3 independent straight outs.

With 3456, you still have a 4 or a 5 to make a better straight (hence 4 outs), but the spades are completely independent of that, so that gives you 1 extra out against the AA hand, plus your defensive 3.

Hmmmm... I think that may mean that either 3456 is the best possible hand, or else there really isn't one, because you get 3456 as favorite against certain boats, but certain boats as favorite against 4567, which is in turn a favorite over 3456 (I'll have to check into this one further).

While I agree in theory with your #'s Aisthesis, I do disagree in a little in play. The redraw to the 7 or 6 high flush is not really that strong unless you can get all your monye in on the flop. If you do get all your monet in on the flop there is a good chance that someone else had 45 which takes away two or your redraw outs. In a regular hand I would much rather have AA45ds. As an example lets say I have 4567 hand it checks to me on the button, I bet the pot, and get 1 or 2 callers I really am not happy making a flush and now after a great flop I can be drawing dead to a bigger flush. At least wth AA45 it might not get there as often but it makes close to the nuts if you improve and getting counterfit sucks with a 4 or 5 but at least I have a redraw.

Definitely true from a practical standpoint in both cases. The Ks is also interesting because a hand like AAKsXs also can't go away (I'm pretty sure) and is pretty easy to play (make it AAKs5s just to eliminate straight flushes).

I'm still interested in figuring it out with open cards--in which case, I almost think 4567ds is the best. It's just that both redraws are unbettable in practice if you make them. Also AA45 with club redraw is easy to play because the club redraw is also to the nuts.

The turn play with open cards might also bring out some interesting aspects (but also allows you to bet non-nut flushes).

Anyhow, Goop, you're right as far as I can tell that there's no hand that's best regardless of what your opponent has. Here's my summary of the hands we've put in play so far:

AcAh4s5c is slightly behind to 3s4c5s6c:
Tie: 26.59%
Win: 35.61%
Loss: 37.80%

AcAh4s5c is slightly ahead to 4c5s6s7c:
Tie: 25.61%
Win: 39.51%
Loss: 34.88%
From a practical standpoint (without open cards), I think it's likely that both players with these hands are going to re-raise at every opportunity (precisely because the ds wrap won't know where he's at with either flush). So, we're all-in on the flop--or is it better for the AA45 at some point to call and see whether the turn helps or hurts? I rather doubt it simply because boating chances actually improve on the river (again, might be interesting to play through with open cards), but of course, any club or non-club is going to help or hurt the AA hand, too.

4s5c6s7s is WAY ahead of 3s4c5s6c:
Tie: 61.59%
Win: 36.46%
Loss: 1.59%
This matchup is so extreme that with open cards, the lower wrap would clearly have to fold to the first raise.

So, without considering how the K-high spade flushes play in here, with open cards, I think one has to put 4567ds as the best hand. It's a slight dog to AA45 with club redraw but a HUGE favorite over 3456ds, to which the AA45 hand is a slight underdog.

Other interesting hands to run in the "best or near best" category: K456ds (or even just with spade draw), AcAh/dKs5s (even though there's no nut made hand--this one is also easy to play). 2345ds is also an interesting one, as well as high wraps with no spades (one good thing about 4567ds is that if you make your bigger straight, you at least didn't get overflushed--with a few exceptions).