The exchange took place as Musk, who convinced Twitter this week to accept a $44 billion buyout, has been garnering headlines tweeting about politics and making the social media platform more fun.
"Tired of having to collectively stress about what explosion of hate crimes is happening bc some billionaire with an ego problem unilaterally controls a massive communication platform and skews it because Tucker Carlson or Peter Thiel took him to dinner and made him feel special," Ocasio-Cortez said in a tweet.
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In response, Musk told the New Yorker to "stop hitting on me, I'm really shy." He also included a blushing and smiling emoji.
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This follows the "Squad" member's response to criticism over a photo obtained by National Review in late December, showing Ocasio-Cortez in Florida enjoying a cocktail without a mask while on vacation with her boyfriend in Miami Beach.
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“If Republicans are mad they can’t date me they can just say that instead of projecting their sexual frustrations onto my boyfriend’s feet. Ya creepy weirdos," she said in a tweet.
In a since-deleted tweet, Ocasio-Cortez responded to Musk and said the billionaire she was directed to was Facebook founder Mark Zuckerberg. Some users posted a screenshot of the exchange before it was removed.
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Musk, who claimed he has gradually shifted more conservative in response to liberals going too far left, jokingly tweeted this week that his next acquisition will be Coca-Cola so he can put the cocaine back into the soda.
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Musk, who made the shock proposal to Twitter less than two weeks ago, said Twitter had "tremendous potential" that he would unlock. He also called for a series of changes, from alleviating its content restrictions to eradicating fake accounts.
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The firm initially rebuffed Musk's bid, but it will now ask shareholders to vote to approve the deal.
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Musk is the world's richest person, according to Forbes magazine, with an estimated net worth of $273.6bn mostly due to his shareholding in electric vehicle maker Tesla which he runs. He also leads the aerospace firm SpaceX.
"Free speech is the bedrock of a functioning democracy, and Twitter is the digital town square where matters vital to the future of humanity are debated," Musk said in a statement announcing the deal.
"I also want to make Twitter better than ever by enhancing the product with new features, making the algorithms open source to increase trust, defeating the spambots, and authenticating all humans," he added.
"Twitter has tremendous potential - I look forward to working with the company and the community of users to unlock it."
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