Much has been made of the need for Jeremy Corbyn to listen to pro-Brexit voters in Labour’s northern heartlands. But it is becoming apparent that his other major support base – young people in metropolitan areas – can no longer be taken for granted.

Several of Labour’s gains in the 2017 election were in university towns and cities where students turned out in droves to support the leftwing leader. On Saturday, however, many of those same young people were demonstrating to show that Corbyn may not be able to count on their support if he doesn’t oppose Brexit. They have a new message for him: “If you’re with us, we’re with you” – the implication being that, if you’re not, we’re not.

The slogan was emblazoned across an enormous red banner in Leeds city centre, where young campaigners gathered to collect signatures for a petition asking Corbyn to back a public vote. Wrapped up in gloves and scarves, with 70s and 80s hits blasting from their portable speaker, they set out to approach the shoppers going about their day.

Among the activists was Ryan Simms, 26, who works in procurement for the NHS in Leeds and has been a Labour supporter for six years – but only joined the party after Corbyn became leader. “I liked him because he wasn’t afraid to be different to the Tories,” Simms said. “Unlike other politicians, he has always stuck to his beliefs, even when they were unpopular. But if he keeps sitting on the fence on this issue, it’ll make me wonder whether he’s not so principled after all.”

Charlie Roberts, 21, and Catherine Fairbairn, 19, who are both students at Leeds, said they would vote Green next time unless Labour backed a public vote. “Corbyn wants an election, but it’ll be one where we have the choice between a Tory Brexit deal and some magical unicorn Brexit deal promised by Labour,” Roberts said.

Emily Dean, 20, also studying at Leeds, is a constituent in Leeds North West, one of the marginal seats gained by Labour in 2017. She usually supports Labour, but next time she’ll vote Green unless Corbyn backs a people’s vote. “I feel like he’s dead against it, even though that’s not the view of most the MPs and members of his party,” she said.

More young activists were out demonstrating in areas including Bristol, Leicester, Warwick, York and Edinburgh. Two of Labour’s biggest target seats are in Edinburgh, held narrowly by the people’s vote-backing SNP. Andrew Wilson, the chair of Edinburgh Labour Students, is coordinating the demo there. “Jeremy Corbyn has a choice,” he said. “He can enable a Tory Brexit or back a public vote. If he allows any form of Tory Brexit to pass, he could lose the young, and could lose Scotland for good.”

When it comes to enabling the Tories, the Liberal Democrats might serve as a cautionary tale for Corbyn. They won 53 seats in 2010 as Nick Clegg’s popularity surged among younger voters in the run-up to the election. But almost immediately after entering government, Clegg reneged on his promise not to raise tuition fees – and his party has not yet recovered from the effects of his decision.

In an open letter to Corbyn, another Labour activist, Cathleen Clarke, who joined the party to back Corbyn in the second leadership race and has campaigned for him in Plymouth, where Labour has recently made gains, said: “If you can’t agree with the arguments, at least appreciate the numbers. The large majority of Labour members want a public vote, with 88% saying they would vote to stay in the EU if there was a public vote.” Imagine the proportion of young Labour members who want to stay – 90, 95%?

“Young people are the generation most in favour of a public vote, and most desperate to stay. Of the 1.6 million young people who have joined the electorate since 2016, who say they are certain to vote in a future EU referendum, 87% would vote to stay in the EU.

“We beat the expectations of the establishment in 2017 because they didn’t pick up my generation’s appetite and enthusiasm for change. Fed up of being beaten down and beaten on, you gave us hope of a better tomorrow. We achieved historic victories in Canterbury, Bristol, Leeds, Sheffield, and Brighton. Yes, we voted for a manifesto that said we would deliver Brexit, and many of us didn’t oppose that then. But that was when a better Brexit deal was possible, and two years before exit day.

She added: “Theresa May is wrong: things have changed. The choice the prime minister thinks she has left us with is her awful deal, or a slightly amended version of it, or no deal. Both of these options sell out our future and must be rejected.

“We must resort to our own backstop we created at Labour party conference – a public vote.”