David Jackson

USA TODAY

COLUMBIA, S.C. — Depending on which campaign is talking, the Republican presidential race is either a three-person, two-person, or one-person affair.

In any event, Donald Trump, Ted Cruz, and Marco Rubio all claimed success in South Carolina and moved Sunday to take the fight to Nevada and other states that will help decide the matter.

"I think we're going to do very well," said Trump, the unquestioned Republican front-runner after a comfortable win in South Carolina, 11 days after an easy victory in New Hampshire. The New York businessman had predicted he would "run the table" after a victory in South Carolina.

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Trump stopped short of declaring ultimate victory Sunday telling CNN's State of the Union that there are "talented people" remaining in the field. "We'll see what happens — certainly nobody's unstoppable," Trump told CNN, though he later added that, "so far, I'm really on my way" and "I'm going to win."

Cruz noted he is the only candidate to have beaten Trump — in the Iowa caucuses Feb. 1 — and said that more and more conservatives realize he is the only who can best him. "That's why conservatives are uniting behind our campaign," Cruz told CBS's Face The Nation.

Rubio, who finished second in South Carolina by a razor-thin margin over Cruz, pronounced it a "three-person race," and the Florida senator predicted traditional Republicans would rally around him in the face of challenges by anti-establishment "outsider" candidates Trump and Cruz.

The Rubio campaign also began overtures to supporters of Former Florida Gov. Jeb Bush, once a Rubio patron, who dropped out of the race Saturday night.

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"We feel like a lot of the people that were on Jeb's team are people we're gonna welcome onto our team," Rubio told reporters aboard a plane to Tennessee, where he made a campaign appearance Sunday.

Up to now, the anti-Trump vote has been split among multiple candidates, Rubio said on CBS' Face The Nation. That will change as more and more rivals drop out, he added, and "we feel really good about this coalescing."

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While Trump and Cruz went after each other viciously in South Carolina, Trump and Rubio have pretty much left each other alone — though there are signs that will change soon.

Trump has called Rubio weak on illegal immigration and re-tweeted a supporter's claim that that the Floridian is also ineligible for the presidency, despite the fact that Rubio was born in Miami.

Asked about Rubio's eligibility on ABC's 'This Week,' Trump said: "I don't know ... I've never looked at it."

Responding on the same program, Rubio said of Trump: "This is a game he plays. He says something that's edgy and outrageous and then the media flocks and covers that and then no one else can get any coverage on anything else."

Two other Republican candidates who finished low in South Carolina — Ohio Gov. John Kasich and retired neurosurgeon Ben Carson — vowed to stay in the race.

“People want to consolidate, they ought to consolidate my way,” Kasich said on Face the Nation.

Next up for Republicans is Nevada, which hosts GOP caucuses on Tuesday after handing Hillary Clinton a win in Democratic caucuses Saturday. Early polls give Trump a huge lead in that state.

Also on the horizon: March 1 contests in a dozen states, mostly southern states that somewhat mirror South Carolina, including Texas, Georgia, Alabama, Tennessee, and Virginia.

The prospect of "Super Tuesday" explains why Trump has an event Sunday in Atlanta, and why Rubio plans to campaign in Tennessee and Arkansas. Cruz has campaign appearances in Nevada.

In bidding to nail down the Republican nomination, Trump has a chance to defeat his nearest rivals in their home states: Cruz in Texas on March 1 and Rubio in Florida on March 15.

Kasich, meanwhile, focused on two other March 1 states, Vermont and Massachusetts. The Ohio governor also plans to make major bids on friendlier terrain in the Midwest, including contests in Michigan on March 8 and his home state on March 15

Trump's opponents have long contended that their numbers will rise as other Republican candidates leave the race. They cite intense anti-Trump feeling within the party because of his critical comments about Mexicans, women, former President George W. Bush, and the GOP establishment in general.

The maverick billionaire disputed that analysis, telling supporters during his victory rally in South Carolina that "as people drop out, I'm going to get a lot of those votes also."

Whoever draws votes from whom, the remaining Republican face figures to be as intense as the contest in South Carolina.

Trump and Cruz accused each other of lying and questioned the other's basic stability. Trump has even threatened to sue Cruz over his eligibility to be the president, citing his birth in Canada. Cruz notes his mother was a U.S. citizen.

Cruz and Rubio also argued about their positions on immigration, a topic that also triggered disputes between Rubio and Trump.

Speaking to backers in South Carolina, Trump signaled the coming fury by discussing the difficulties of running for president.

"It's tough, it's nasty, it's mean, it's vicious — it's beautiful," Trump said. "When you win, it's beautiful."