MIAMI – “I’ve been working and fighting against the communists since I was 11 years old,” said Emilio Izquierdo, a 71-year-old Cuban-American exile, as he stood outside the legendary Versailles restaurant in the sweltering heat of Little Havana.

For decades, people like Izquierdo have come to this place to see friends, eat authentic Cuban food and talk politics. On Wednesday, as patrons ordered Cuban sandwiches and café con leche, Izquierdo was getting fired up talking about the rise in socialism in the United States -- something, he said in an interview with Fox News, he fears will “destroy America.”

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The flirtation by Democratic presidential candidates with socialistic policies comes as polling indicates Democrats could have an opening to chip away at the traditionally solid GOP support from the Cuban-American community, particularly with younger voters. But the older generation, in conversations at Versailles, suggested Democrats could be alienating the community by openly embracing socialistic policies, as candidates did for two nights in Miami this week.

“They don’t like the socialism,” 81-year-old Antonio Roque, who was born in Cuba and came to America in 1962, said of the Cuban-American community while standing at the counter of Versailles, amid a steady stream of tourists and locals. “I don’t like it.”

Another patron, 53-year-old Gonzalo Lopez, who was also born in Cuba, said he decided against watching the Democratic debates because “I don’t like any of the Democrats right now…I don’t like socialism.”

“It’s a fight between democracy and socialism,” Lopez said. “I’ve got nothing to do with the second one.”

Ahead of the debates, the Republican National Committee visited Versailles in an effort to tie Democrats to socialism -- and keep their grip on the voters there.

“There is no community that understands that better than the Cuban community here in Miami, many of them fled the Castro regime because they know what socialism means,” Republican National Committee chairman Ronna McDaniel said during her visit Wednesday.

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The national Democratic Party has repeatedly pushed back against Republican claims that they are the party of socialism now. DNC Chairman Tom Perez recently said on "Fox News Sunday": “This is one of the oldest tricks in the playbook. You go back 75 years, when Republicans don’t want to discuss the issues that matter to real people, they call it socialism.”

But during the debates on Wednesday and Thursday night, many of the candidates embraced socialistic policies, like "Medicare for all," free public college and a 70 percent tax rate for the wealthy.

And in an embarrassing move that caused a backlash, New York Mayor Bill de Blasio on Thursday quoted Marxist revolutionary Che Guevara to pump up a crowd of union members while campaigning in Miami. "Hasta la victoria, siempre!" de Blasio shouted. The phrase translates to "Until victory, always!" and is attributed to Guevara, a key figure in the Cuban Revolution led by Fidel Castro. (de Blasio later issued an apology, saying he didn't know the phrase was connected to Guevara.)

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Several of the Democrats on Thursday night pushed back against the embrace of socialism, with former Gov. John Hickenlooper of Colorado arguing it’s not good politics.

“I think that the bottom line is if we don't clearly define we are not socialists, the Republicans are going to come at us every way we can and call us socialists,” Hickenlooper said.

That's just what they were doing at Versailles this week.

As she was touring the restaurant on Wednesday, McDaniel stopped to talk with Izquierdo, who was wearing a “Trump 2020” hat. He told the RNC chairwoman that he doesn’t identify as Republican or Democrat, but would vote for Trump in 2020. When McDaniel said she liked his hat, he offered to let her keep it.

“I’m going to show it to the president,” she said.

Fox News’ Brooke Singman and Nick Givas contributed to this report.