Pedophile Story? (possible spoilers)

Well it's not pedophilia since he is not a child but a teen. It is wrong obviously, an adult should know better and respect the law. But I am more trouble by all the gay antics from the two main actors, hugging each other, doing all this gay stuff in their promo, obviously pandering to the new fascist liberalism, yet they are not gay, they haven't had to go through the risks, the bullying, the danger. It's almost like if it was a celebrated minstrel show. Why do THEY get to do it? If it was two gay actors or one gay actor it would be frowned upon. It also begs the question, why weren't gay actors cast on the roles? The whole situation is gross to me. And the movie looks like shit.

In a lot of places in Europe the age of consent is 14.

I read that the older character is 24, not 30..
Sorry but I don't see any problem with this be it homosexual or heterosexual..It is not pedophilia...as far too many incorrectly label and report.
Geez..people need to truly learn what certain words actually mean...

And to be honest, that sort of age difference is common in real life. Or it was for me..
I experienced, as a heterosexual female, several older romances as a teen and I did know the difference between that and predator. I don't consider 17 and 24 that worrisome. It depends on the situation
The biggest problem would be if one were in certain power positions; like a teacher.

But people need to stop being offended by everything. It's getting old.
If something doesn't work for me I just won't watch it..Everyone, these days...is offended by everything as if the world just turns for them.
I could care less about this movie in particular because even heterosexual romances tend to bore me..so homosexual coming of age stories likely just won't do much for me and my interest..

Your sensationalistic thread title is inaccurate. You see, Oliver's relationship with Elio would have made him a ephebophile having a primary sexual interest in mid-to-late adolescents, generally ages 15 to 19 and not a pedophile. In addition, Oliver was an upcoming professor who had no homosexual leanings before he met Elio. The age of consent in most countries is 16 and 14 in many parts of Europe. There was a very special connection and bond between the two.

Here is how I look at it. First, the ages are 17 and 24 (and in real life about 20 and 29 while filming). The age difference is definitely there for a reason, but not for the point of him being "underage" and it being predatory, but more so to showcase the contrasts as well as similarities between where they both are at in life. That is expressed in terms of being comfortable with themselves and who they are as a person and who they want to be with and how and why they end up bonding. The ages are to help better contextualize the background, family, culture, era, intelligence, confidence, experience, maturity, self-awareness, etc. of the characters. This also helps explain the reality of the relationship in terms of if it can last or how it can last and how they are both going to feel about it. If it makes you uncomfortable act like the younger character (Elio) is 18 or 19 since the point isn't about age but rather the stage in life. Elio is at an age where it's his first love and obsession and different sexual feelings are coming to the surface and having him younger is a realistic stage in life when that would happen. There is the argument too that age shouldn't matter because its legal and the age of consent in Italy even younger than 17, which maybe you don't think is right but that what it is. There is also the argument that Elio is mature for his age and they are intellectually on the same level. Elio is also is pursuing the older character and its not predatory at all and it is all consensual. I don't really like the general argument or excuse of "well he's mature for his age" because thats a common creep move in real life for excusing predatory behavior, but the truth is it is all legal and consensual and the exact age isn't the point. Its not a pedophile story in legal or thematic terms. ( I am not very good at writing or consolidating my thoughts so I hope this made at least a little sense)

I recommend researching ephebophilia and hebephilia, both of which, like pedophilia, are chronophilias but are indeed not pedophelia. Were Armie Hammer's character a man who exclusively pursued relationships with boys the age of Timothee Chalamet's, he may be an ephebophile. However, if his attraction isn't exclusively to boys this age, he wouldn't be.

All I know about life for gay boys and men I know from reading and listening, so I can't authoritatively speak to their experiences, but I understand that for many gay men relationships with older men is a big part of coming of age and into one's own in a world that even in 2018 is shunned by so many.

What is the age of consent wherever they are in Italy?

The age of consent in a large portion of the US is 16 or 17. In the rest of the world, sometimes lower. Some 17 year olds are wiser and more mature than some 30 year olds. It would only be morally questionable if one party was in a role of authority over the other, or if there was a large disparity in the mental capacity or maturity level of the two parties. Not having seen the movie, I don't know if that was the case.