The only reason why Vito Corleone has to see Sollozzo is that the Tattaglias ask him to do it.
Now, when Vito mentions the Tattaglias, Sollozzo gives "My compliments" to Tom, which obviously refers to Tom's ability to find out about this connection.

Thoughts?

The film portrays that scene differently than the novel. In the novel, Vito already knows that Sollozzo is vouched for by the Tattaglias. I think that's left out of the film.

I'm not sure why the compliment dialogue takes place. Perhaps it's to illustrate Vito's depth of knowledge about NY underworld activity and to highlight the fact that Sollozzo didn't volunteer his affiliation with the Tattaglias.

In the film, there's nothing to indicate that the Tattaglias ask Vito to see Sollozzo. There's only Tom saying that Vito will have to see him and Vito's references to Sollozzo as a "serious man" to be "treated with respect" indicating some Mafia status.

I've always believed that Sollozzo pretended to hide his connection to the Tattaglias from the Corleone Family. That way, once Tom "discovered" the connection, he would not be inclined to dig any deeper and Barzini's role would remain hidden.

I completely agree, Mr. Woltz. Well put.

Logically, Sol never would have gotten an appointment with Vito to discuss big business without a recommendation from a major family--otherwise, he'd be just some guy off the street. I think Sol was buttering up Tom. He already surmised that Tom would go for his deal. So, by flattering him with "my compliments," he'd be doing more (in his mind) to get Tom on his side.

I agree w TB that Sol was just schmoozing Tom. No question he had to come highly recommended or vito would not have met him. I thought Tom really distingiushed himself by all the digging he did into the Turk's background to brief Vito. He was right about everything right down to his being "good with a knife" in mattters of business.

If Tom were REALLY good, he'd know what the Tattaglias' piece of the action was going to be.

Surely Sollozzo was recommended or at least connected and well known. Tom mentions to Vito that "Virgil Sollozzo called...we're going to have to give him a day next week".
To me, this implies that Sollozzo can call the don (or at least Tom) on his own. Tom's notes imply that he got some information from Italy (Tomassino?) and some from the U.S. Hell, for all we know the Corleones have their own spies in the Tattaglia family.

Yes, Sollozzo was schmoozing Tom and presenting the illusion that his side deal with the Tattaglias was found out. And by offering his compliments, he hopes that will be enough for them to not dig any further.

But Vito, being the intuitive man he was, knew there was something more to Sollozzo's deal which is why he sent in Luca Brasi.

I don't think it was intuitive of Vito to call in Luca. He recognized immediately that Sonny's very public gaffe was potentially lethal. Vito's gaffe was to send Luca to the Tattaglias with that ridiculous story that he was "not too happy with our family."

I always took the "look under his fingernails" to mean that there was a big chunk missing from that meeting. I mean, you were there, right? You could tell Sollozzo was hiding something. Vito needed to know what that something was.
Sonny was paying attention to what Sollozzo was saying.
Vito was paying attention to what Sollozzo wasn't saying.