Twitter and its chief executive, Jack Dorsey, placed warnings on three of President Trump’s tweets last week, taking a measured but hotly debated step to place some limit on the president’s use of social media to spread falsehoods and incite his followers.

Twitter attached labels refuting two of Mr. Trump’s tweets on voter fraud and restricted one that implied protesters in Minneapolis could be shot. But it left countless others unchallenged, including those baselessly insinuating that the MSNBC host Joe Scarborough killed a former staff member.

A New York Times review of the president’s 139 Twitter posts from Sunday, May 24, to Saturday, May 30, found at least 26 contained clearly false claims, including five about mail-in voting that were not flagged, five promoting the false conspiracy theory about Mr. Scarborough and three about Twitter itself. Another 24 were misleading, lacked context or traded in innuendo. (This analysis did not include dozens of Mr. Trump’s retweets.)

To put it another way, more than a third of the president’s tweets over the course of a week contained dubious information. That presents a challenge both to Twitter and to the millions of people who are exposed to Mr. Trump on social media, especially now, with the nation facing the triple challenge of a pandemic, economic dislocation and nationwide protests over systemic racism.

Unsubstantiated charges of fraud in mail-in voting

Twitter attached information to refute two of Mr. Trump’s posts about mail-in voting that falsely claimed that California was sending ballots to “anyone living in the state no matter who they are or how they got there.” State officials will mail ballots to registered voters only.

Mr. Dorsey said that those tweets on May 26 specifically violated the company’s civic integrity policy as they “may mislead people into thinking they don’t need to register to get a ballot.”

Five others posts by Mr. Trump repeated his general falsehoods about mail-in voting but did not specify any one state’s election process or ballot distribution plans — and were not affixed with a label.

Two days before Twitter labeled the two tweets on May 26, the president warned twice that the upcoming presidential election will be “rigged” through mail-in ballots. After Twitter’s actions, Mr. Trump continued to claim that mail-in voting would lead to a “free for all on cheating, forgery and the theft of Ballots,” a ”tainted” election process, “MASSIVE FRAUD AND ABUSE” and “THE END OF OUR GREAT REPUBLICAN PARTY.”

There is no evidence for any of these claims. Voter fraud in general is extremely rare. While mail-in voting is less secure than in-person voting, fraud incidence rates remain extremely low — one study found an improper voting rate of 0.004 percent. Studies have found little evidence that mail-in voting and so-called no-excuse absentee voting benefit one political party over another.

Inaccurate claims about Twitter

Mr. Trump responded to Twitter’s actions by issuing an executive order that seeks to strip liability protection in certain cases for companies like Twitter, Google and Facebook for the content on their sites. If carried out, the order could lead to the companies facing legal liability for false and defamatory statements posted on their sites.

The president also took to Twitter to castigate the platform in three false posts.

He accused the company of “completely stifling FREE SPEECH.” But this is a misreading of the First Amendment, which prohibits Congress from “abridging the freedom of speech.” While the Supreme Court has held that this applies to all government agencies, it does not apply to private companies like Twitter.

In two tweets on Friday morning, Mr. Trump complained that Twitter had not flagged “China’s propaganda.” But the company had attached similar warning labels two days earlier to tweets posted by Zhao Lijian, a spokesman for China’s foreign ministry, that suggested that the coronavirus originated in the United States.

Baseless claims about ongoing protests

Mr. Trump has dismissed the nationwide protests over the killing of George Floyd by suggesting they are not organic or legitimate.

In three tweets, he said without evidence that the protests across the country and in front of the White House were “professionally organized” and “have nothing to do with George Floyd.”

While it is impossible to know the motivation of every person participating in these demonstrations, Mr. Trump’s broad generalization discounts the thousands who have taken to the streets specifically to protest the killing of Mr. Floyd, a black man who died after being handcuffed and pinned to the ground by a white police officer.

In another tweet on Saturday, Mr. Trump said that “80% of the RIOTERS in Minneapolis last night were from OUT OF STATE.” This was an estimate first offered by Gov. Tim Walz of Minnesota, a Democrat. But after reporting by local news outlets that suggested the opposite was true, Mr. Walz declined to repeat the estimate and said that more data was needed to properly characterize the proportion of those arrested who were not local residents.

These theories echoed Mr. Trump’s previous claims that those who protested the confirmation hearings of Justice Brett M. Kavanaugh in 2018 were paid by George Soros, the billionaire investor and Democratic donor.

Mr. Trump did not name the supposed organizers of the current protests.

Other falsehoods

Other inaccuracies from a week of Mr. Trump’s tweets centered on familiar foes and oft-repeated boasts.

Mr. Trump tweeted, with no evidence, that Speaker Nancy Pelosi had complained that he had moved too quickly in imposing some restrictions on travel from China in response to the coronavirus. The Times was unable to find an instance of Ms. Pelosi publicly addressing or criticizing that decision.

He repeated his claim that former Vice President Joseph R. Biden Jr., the presumptive Democratic presidential nominee, had “apologized” for opposing the policy. No record of an apology exists.

He falsely accused Representative Conor Lamb, Democrat of Pennsylvania, of breaking his campaign promise to vote against Ms. Pelosi for speaker. Mr. Lamb, who won his seat in 2018, voted for Representative Joseph P. Kennedy III, Democrat of Massachusetts, when the House held its election for speaker last year.

He falsely claimed on Saturday that Washington’s Democratic mayor, Muriel E. Bowser, “wouldn’t let the D.C. Police get involved” in monitoring protests outside of the White House on Friday. But the Secret Service, which Mr. Trump praised in the same tweet, said in a statement that it had made six arrests that night and “the Metropolitan Police Department and the U.S. Park Police were on the scene.”

He again took undue credit for the Veterans Choice health care program. The program was created in 2014, developed by Senators John McCain and Bernie Sanders and signed by former President Barack Obama — three of Mr. Trump’s political enemies.

Mr. Trump’s declaration that “it was me who shattered 100% of the ISIS Caliphate” was also not true. About a third to a half of the territory formerly held by the Islamic State was regained under Mr. Obama’s administration, according to military and independent estimates. And officials and experts had always anticipated that the campaign, started in 2014, long before Mr. Trump took office, would result in pushing the extremist group out of its self-declared caliphate.

Half truths and murky accusations

Three dozen tweets from the president occupied a factual gray zone. Some were typical examples of political spin, neither completely true nor totally wrong.

Twice, he claimed to have banned travel from China and to have done so “before anybody thought necessary” to contain the spread of the coronavirus. These were exaggerations. The restrictions did not amount to full ban. They did not apply to American citizens or green card holders, and they contained other exemptions. Numerous other countries had taken similar actions before Mr. Trump did.

He misleadingly boasted of the United States having carried out 15 million coronavirus tests, “by far the most in the World,” and the number of cases and deaths “going down all over the Country.” The raw number of tests, while accurate, did not reflect that the United States continues to lag other countries in testing per capita. Cases and deaths were decreasing across the country as a whole, but not in some states.

Ten tweets were devoted to the announcement of grants to local transit agencies from the Department of Transportation. Left unsaid was that these grants have been routinely awarded since the 2013 federal fiscal year and in the first two years of his presidency, Mr. Trump’s proposed budgets called for the grants to be phased out.

Other tweets were ambiguously worded, making them difficult to fact check even as they hint at nefarious activity.

Mr. Trump twice said that social media companies “attempted”and “failed” to do something in the 2016 election, but never specified what exactly the companies attempted. (In the past, he has mischaracterized research to mount a baseless suggestion that Google “manipulated” votes.)

Perhaps there’s no better example of how Mr. Trump trades in vague claims than his repeated allegations of the “greatest political crime” or scandal in history, committed by the Obama administration to undercut his 2016 campaign and the start of his presidency. In four tweets, the president echoed this but never specified what that crime was. In others, he simply referred to “Obamagate.”

Curious about the accuracy of a claim? Email factcheck@nytimes.com.