Tom Riddle - who was trying not to think about the fact that he currently occupied the body of a pre-pubescent girl and the numerous reasons doing so was VERY WRONG (kinda like that game where you try not to think about pink elephants, and end up constantly thinking about pink elephants) - had been walking towards Moaning Myrtle's bathroom which contained the entrance to the Chamber of Secrets when he saw the notice written under the words he had put up to foster fear and widespread panic. After blinking in shock and confusion, finally processing what he was seeing, getting into a towering rage, swearing until the air around him was literally blue, and finally calming down, he transfigured a couple rolls of toilet paper into a pot of red paint and a paintbrush he posted his response to the claim of heirship made by someone who couldn't possibly back it up.

WHAT RIGHT DO YOU HAVE TO THE TITLE YOU CLAIM? I AM A DESCENDANT OF MARVOLO GAUNT AND THE TRUE HEIR OF SLYTHERIN!

The next day, Tom, aka Voldemort got his answer. Written in Harry Potter's messy scrawl was the following:

As the only surviving magical descendant of Marvolo Gaunt's son and eldest child Morfin's ONLY child, I am the heir of Slytherin. My muggle cousin might be able to make a better claim considering the fact that he is the son of the eldest daughter of Morfin's daughter.

Tom's reply appeared a week later. It was in red Glow-in-the-dark poster paint in letters 18 inches high.

MORFIN'S DAUGHTER WAS A BASTARD. AS A DESCENDANT OF A BASTARD, YOU HAVE ABSOLUTELY NO RIGHT TO THE TITLE YOU LITTLE SHIT!

Harry's response appeared magically overnight in a rather fetching shade of crimson (rather Flitwick had been amused by the exchange between the "Heirs of Slytherin" and had noticed that the argument had reduced some of the tension that had been building since Halloween, and decided to ignore the paintbrush that seemed to be moving by itself and the floating tin of paint that was leaving spots on what was obviously an invisibility cloak worn by one Harry Potter who was out after curfew) . Beneath Tom's mini-rant on the legitimacy or lack thereof of Harry's line was:

Actually, precedent was set when the illegitimate grandson of the former head of the Black family was given the title of Lord of the Noble House of Black in 1357. A title which - like all titles the so-called Purebloods hold - is completely meaningless outside the wizarding world. The only reason I've even bothered claiming the stupid title of "Heir of Slytherin" is because you've been misusing property that is mine according to wizarding law, asshat.

A week and a half later the response that appeared left Harry wondering if the other "Heir of Slytherin" was severely brain damaged, and Tom wondering if having an eleven year old girl pour her soul into him was a good thing. For in the now familiar glow-in-the-dark paint that had given Filch fits was Tom's reply.

I KNOW WHERE THE CHAMBER OF SECRETS IS AND YOU DON'T. FINDERS KEEPERS, LOSERS WEEPERS. SO THERE.

A day later, Harry's response appeared. In a rather nice shade of Vermilion were the words:

For some strange reason I thought you might be Malfoy, but I was mistaken. Despite the fact that his constant attempts to provoke me are exceedingly juvenile, he would never stoop to the level of stupidity you have.

From there, until the day Tom walked Ginny down to the chamber of Secrets, (other than the memorable occasion when Harry got a lawyer to file a restraining order barring the "False Claimant to the title of Heir of Slytherin" from coming within a hundred feet of the Chamber of Secrets, and Tom painted over his copy of it which was posted beneath the word Wanker with an entire bottle of red glitter nail polish) the argument had pretty much devolved into name calling.

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Author's Note: And there you have it, the much requested argument between Tom and Harry. I have the feeling that it most likely wasn't as funny as the one you were picturing in your heads.