Student protests against Israel’s war in Gaza continue to spread across the US, following last week’s arrest of more than 100 demonstrators at Columbia University.

There have been nearly 550 protest-related arrests in the past week at major US universities, according to a tally by news agency Reuters.

The students want universities to cut ties with companies helping Israel’s war in Gaza and, in some cases, with Israel itself.

Some universities have called in police to end the demonstrations, resulting in clashes and arrests, while others appear to be biding their time as the academic semester enters its final days.

The University of Southern California cancelled its main graduation ceremony, set for 10 May, after the arrests of 93 people at the Los Angeles campus on Wednesday.

At Boston’s Emerson College, 108 people were arrested overnight with video showing students linking arms to resist officers, who then moved forcefully through the crowd, throwing some students to the ground.

Student protester Ocean Muir said: “There were just more cops on all sides.

“It felt like we were being slowly pushed in and crushed.”

She said police lifted her by her arms and legs to carry her away and she was charged with trespassing and disorderly conduct.

At Emory University’s Atlanta campus, 28 people were detained and the local branch of activist group Jewish Voice For Peace said police used tear gas and tasers on protesters.

Police there admitted using “chemical irritants” but denied using rubber bullets.

Cheryl Elliott, Emory’s vice president for public safety, said the aim was to clear the area of a “disruptive encampment while holding individuals accountable to the law” but human rights groups questioned the “apparent use of excessive force” against free speech.

Charges were dropped, meanwhile, against 46 of the 60 people detained by police at the University of Texas.

At Indiana University Bloomington, police with shields and batons shoved into a line of protesters, arresting 33 people.

At City College of New York, police officers retreated from protests, to cheers from the hundreds of students gathered on the lawn on the Harlem campus.

At California State Polytechnic University in Humboldt, students have been barricaded in a campus building since Monday, with staff trying to negotiate.

At University of Connecticut one protester was arrested and tents torn down, while protests continued at Stanford University and the New Jersey campus of Princeton University.

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Harvard University is among those that have not taken action against protesters who have set up tents.

At New York’s Columbia University, where the protest movement began, university officials remain locked in a stalemate with students.

Police cleared tents and arrested more than 100 people last week but students put the tents up again in an area where graduation ceremonies will be held in a few weeks.

The administration has given protesters until Friday to leave.

There have been accusations that some pro-Palestinian protesters have harassed or abused Jewish students but protesters blame outsiders trying to infiltrate and malign their movement.

Protest leaders admit there has been abuse directed at Jewish students but insist the protests are not antisemitic.

Some of the universities have seen counter-protests from Israel supporters.

Two courts aren’t enough – not for Donald Trump, not on a Thursday.

His hush money criminal trial and Supreme Court hearing, legal events one and two, were already under way when he scored the hat-trick.

A New York judge announced he was upholding the verdict and the $83m damages award against Trump for defaming writer E Jean Carroll.

There’s a lot going on, legally, and there’s a lot riding on all of it for Trump.

Catch up: how the day unfolded in court

The hearing at the Supreme Court concerned the 6 January riots, election subversion and Trump’s alleged involvement. It is a crime against democracy, at the serious end of the legal jeopardy he faces.

His lawyers argued he should be shielded by immunity from prosecution for what he did while acting as president.

The prosecution’s case is that he was acting as a private citizen, not in an official capacity.

Trump wasn’t present at the hearing in Washington DC, but he will have liked what he heard.

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The prevailing legal assessment is that discussions with the nine-judge panel indicate that, while they didn’t necessarily agree with his argument for immunity, they have enough questions to delay the prosecution further.

A majority appear to think that presidents have some immunity from criminal prosecution for their official actions, even if the exact parameters are unclear.

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What is clear is that if the trial court is instructed to determine which of Trump’s allegedly illegal acts qualify for immunity as official acts, it will be an extended process that could easily push the trial beyond the November election.

Such a scenario would suit Trump. The less criminal exposure he has before America votes, the better for him.

If he can push the trial past November, and win back the White House, he can use the power of office to make the charges go away.

The New York hush money trial is the only one of four criminal prosecutions to have begun.

The Supreme Court appears set to shorten the odds on it being the only one before America goes to the polls.

It is the pressing matter of the truth, the whole truth and nothing but the truth about the man who would be president, and it’s a race against time.

This stress test of the fundamentals of American democracy and rule of law gets ever more stressful.

Donald Trump managed a partial victory in the Supreme Court today, as justices delayed any potential decision on his immunity case over election riots.

Trump argued on Thursday he has total immunity over the 2020 riots and while justices in the Supreme Court were not convinced by his arguments, some raised the point he may have some level of immunity – and delayed any potential decision on that until June.

If they then rule the former president does have a level of immunity, it could kick the issue back into lower courts to decide what that level is, and knock back any potential decision to beyond the November election.

On Thursday, Trump, who made history as his country’s first ex-leader to face a criminal trial, was also fighting on two other separate legal fronts. They include:

• His hush money trial in New York where he is accused of falsifying business records after allegedly paying money to porn actress Stormy Daniels to “cover up an affair”.

• His defamation case, brought by writer E Jean Carroll – a judge rejected Trump’s attempt to throw out the verdict against him, leaving him facing an $83.3m (£66.5m) payout.

And adding to Trump’s legal woes, his former lawyers and associates were indicted on Wednesday in a 2020 election-related scheme in Arizona.

Follow latest: Trump’s hush money and Supreme Court trials
Analysis: Trump has won qualified victory at Supreme Court

Supreme Court

During proceedings, justices appeared likely to reject Trump’s claims of total immunity, but delayed any ruling to make a decision over what specific immunity he may or may not have.

Trump, 77, had even asked to skip his New York criminal proceedings to sit in on the Supreme Court’s special sessions.

In Washington, the lawyer representing the special counsel told the court it had never been previously recognised what kind of immunity Trump was actually seeking.

Chief justice John Roberts said he was concerned if presidents were not immune, the country would rely on “good faith” to prevent abusive prosecutions against presidents.

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He told the special counsel’s lawyer, Michael Dreeben: “Now you know how easy it is in many cases for a prosecutor to get a grand jury to bring an indictment.

“And reliance on the good faith of the prosecutor may not be enough in some cases – I’m not suggesting here [Smith’s indictment of Trump].”

The Supreme Court is expected to release its opinions by the end of June over whether Trump has immunity or not.

With five justices appearing likely to reject Trump’s claims of absolute immunity, some suggested the former president may have some level of immunity.

If the eventual ruling reflects that, lower courts may be required to sort out the specifics of this – which could push any eventual decision past the November election.

Hush money

Meanwhile, in New York, Trump was once again present in Manhattan’s criminal court, accused of falsifying business records.

Earlier this week, the court heard from AMI, the former publisher of the National Enquirer, about an alleged “catch and kill” scheme, which was said to have been used to get rid of negative stories about Trump.

David Pecker, boss of AMI who signed a no-prosecution deal to testify, described shelling out hundreds of thousands of dollars to buy up rights to potentially damaging stories.

The National Enquirer, the court heard, bought up a sordid story from a New York City doorman as well as accusations of an extramarital affair with a former Playboy model to stop the claims getting out.

But Mr Pecker reached his breaking point with Stormy Daniels – a porn actress who was allegedly paid by Michael Cohen, Trump’s former lawyer, to keep quiet over her claims of a 2006 sexual encounter with Trump. Something he denies.

Mr Pecker told jurors his publication had been contacted by Ms Daniels’s representatives who said they could buy her story for $120,000 (£96,000) if it decided right away.

However, the publishing boss refused to. He told Mr Cohen: “I am not paying for this story. I didn’t want to be involved in this from the beginning.”

After that, a cross-examination of Mr Pecker began, with one of Trump’s lawyers, Emil Bove, taking centre stage.

Gag order

Hanging over Thursday’s hush money proceedings were allegations that Trump, once again, violated a gag order.

The order restricted Trump’s public speech regarding jurors, potential witnesses and some other individuals involved in the case.

Judge Juan Merchan was already considering whether to hold Trump in contempt and fine him for what prosecutors alleged were 10 separate violations of the order.

But on Thursday the prosecution ticked off fresh instances of alleged breaches.

These were additional remarks made about Mr Cohen, and a comment Trump made about the jury being “95% Democrats”, among other things.

But Trump was previously dismissive about the threat of having to pay up when speaking outside court, saying he had “no idea” whether he would be fined.

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E Jean Carroll

While things may have went his way partly in the Supreme Court, a judge rejected Trump’s attempt to get a defamation verdict against him thrown out.

Writer E Jean Carroll said Trump defamed her after she accused him of raping her decades ago.

The court ordered Trump to pay $83.3m in damages, and on Thursday, US district judge Lewis Kaplan said Trump was not entitled to a new trial or judgement, so had to pay up.