Okay, lets say Hyman was behind the assasination attempt on Michael (which is the popular theory). Michael goes and acts all buddy buddy with Roth, and says that Pentangalie was behind it, and that he would be killed.

Now, another popular theory is that Hyman Roth had the Rosatos go after Pentangalie. Why? If Michael said that he would kill Pentangalie, why would Roth go and kill him? This, as of now, is the one thing I don't understand out of all three films.

No, I didn't search for them...I'm just not in the mood. But, I guess, if you guys wanna be mean I will try and gain the extra effort to search for it...

Because at that point Hyman knew that Michael didn't suspect Frank. So he already thougt Mike wouldn't kill Frank.

And then he thougt...well...if the both go after me I'm gone...so I'll kill one of them.

Well wouldn't this logic only prove to Michael that Hyman did it? If Hyman knew that Michael was lyeing to him in saying it was Frank, then wouldn't he know that Michael really suspected him as the man behind the assasination?

But Hyman didn't knew....only when Frank was sent by Michael to settle the problems with the Rosato Brothers...at that point he knew that Michael knew he was behind it.

Hmmm, so why would Hyman continue with the Cuba deal if he knew that Michael knew that Hyman was behind it? If that makes as much sense as it did in my head...

Michael was acting on his father's advice: "Keep your friends close, and your enemies closer." When he called on Roth and said, "Frank Pentangeli is a dead man," he was making Roth believe he blamed Pentangeli for the Tahoe attack.
BUT: When Michael dispatched Pentangeli to settle his problems with the Rosato brothers, Roth knew that he didn't blame Pentangeli after all--meaning that he might suspect Roth.
So, Roth ordered the Rosatos to kill Pentangeli. Pentangeli was Michael's man in New York City, and still, apparently, his loyal ally. Roth tried to have him whacked before he and Michael could cook up any mischief against Roth. Probaby Roth didn't tell his motive to the Rosatos--he probably said, "Hey, look, this guy's never gonna settle with you the way you want. Get rid of him now. Let me worry about Michael Corleone."
Now, Moscarelli, to your excellent question: If Michael knew that Roth was behind the Tahoe attack--and Roth knew that Michael knew it--why did Roth invite Michael to Havana? In a word: greed. Roth had made a deal with Michael for him to take over Roth's Cuban gambling operation in return for a $2 million "gift for the President" [of Cuba]. Roth wanted that $2 million. He never felt he was in danger because, as the most powerful gringo on the island, he could have his pals in the Cuban government squash Michael like a bug. He figured, "Heh-heh, I'll lure him into my island stronghold, whack him--and make him pay $2 million for the privilege."
Why did Michael go to Cuba, where he would be in absolute, mortal danger? Because he knew that Roth was greedy for the $2 million. He knew that Roth wouldn't move against him--and his family back in Tahoe--unless and until Roth got the $2 million. Meanwhile, by stalling Roth, he was buying time to find out who was the traitor in his family--and, as we saw, he did find out. Roth wasn't worried about Michael because he arrived with only one bodyguard--and what was one bodyguard against all the Cuban military muscle Roth could count on?
Of course Roth underestimated Michael. But then, Michael underestimated Roth before the Tahoe shooting, and did it again at the Senate hearing. One of the recurring themes in the Trilogy is how powerful people underestimate each other.

Wow, thank you Turnbull, and you too Floors for all your help.

As I detailed in another post on another topic, I think that the plotline developed by FFC followed real Mafia events closely. The Rosatos are the Gallo brothers in real life and Michael was a version of Joe Bonanno/Joseph Profaci whom the Gallo brothers were in revolt against. Pentangeli is Joe Maglicco who took over the Profaci family when Joe died. Bonanno and Profaci and Magliocco were allies. So, Gambino et al backed the Gallos against Bonanno.

Accordingly, Roth (Gambino) would do anything he could to harm Michael.

At what point(= due to what incident) did michael conclude with certainity that Roth was behind the attack.

Long answer:
As you know, the first time he states that it was Roth was when he met with Pentangeli in New York.
I believe he was 99 percent certain it was Roth within minutes of the attack. I think he concluded, with his "Sicilian cunning" (a term much-used by Puzo in the novel), that Frankie was too obviously set up to be the fall guy for the shooting. Frankie (not Roth) came to the party. Frankie (not Roth) had a beef with Michael over the Rosato brothers. Frankie shouted at him. Therefore, Frankie had the motive to kill him. But it was too pat, too obvious, too cut-and-dried. And Frankie wasn't smart enough to pull off the attempted hit and arrange for the shooters to be killed before they were found.
Vito had taught Michael to think as those around him think. He may finally have realized that, although Roth and Vito had been partners in the Prohibition-era booze business, they weren't partners in the Western and Cuban gambling empire tht Roth ran. Michael had been muscling in on Roth ever since he got back from Sicily. Michael killed Moe Green, Roth's best friend, to get control of his hotel. Now Michael owned or controlled three Nevada hotel/casinos, was about to muscle into a fourth one partly owned by Roth, and had designs on Roth's Havana operations. So I think that, after the shooting, Michael realized that Roth (who also benefited from Vito's teaching) was keeping Michael, his enemy, close by treating him like a son and promising to make him his heir-apparent, even as he was planning to kill him.
BTW: I believe Michael was 99 percent certain it was Roth when he met with Frankie. He became 100 percent certain when Frankie agreed to "settle [his] problems with the Rosato brothers." By agreeing to the meeting, Frankie put himself in mortal danger--proof that he was still loyal to Michael. And it was a win/win for Michael: If Frankie came to terms with the Rosatos, that was one less worry for Michael. If they attacked Frankie, it was more proof-positive that Roth was behind the Tahoe attack. Clever Michael!

To support what Turnbull is saying, I would add that the scene between Pentangeli and Michael where Pentangeli makes it clear that Vito Corleone did business with Hyman Roth, Respected Hyman Roth but never trusted Hyman Roth goes unchallenged by Michael. Also, later in the film, Michael sarcastically ponts out that Roth has been dying of the same heart attack for forty years. These are deep rooted and long standing suspicions. Also when Michale tells Tom that he, Tom is the only one he can trust he adds "if what's happened is what I think what's happened, I have to leave here tonight. This means he formed a theory that it was Roth andhe had to put his plan into gear.

I agree with both of you Turnbull and Dontomasso that the suspicion was well founded and logical. But didn't michael already have too many enemies? It could be any of the men from the families whose heads he murdered. Or it could be Fredo or any insider's plan to kill michael without any trigger/inducement from Roth. I agree that if the only choices are Roth and Pentangeli, michael was right in suspecting Roth, but were these 2 the only suspects?

Logical question, svsg. I'm sure he considered other choices--briefly. But he probably concluded that Roth had the strongest motivation.

One of the most powerful factors in his decision had to be the ruthlessness of the attack--machine-gunning his bedroom with his wife in it. Showed the enemy was so intent on getting him that he didn't care who got killed with him. And consider the resourcefulness of the enemy in getting his men inside the compound--and then having them killed right after the attack. No mere revenge-seeker could or would do that. As dt pointed out, Michael couldn't dismiss anyone other than Tom as the inside person, but I think he figured that the insider was working for or with Roth, not by himself.

Could the would be assassins been two or three of "Fredo's men" who may have been planted by Ola with Fredo's unwitting consent? I am thinking two shooters and a third who killed them immediately.