By former President Donald J. Trump’s account, the country is awash in crime. But in fact, under President Biden, the rate of violent crime has fallen.

It is a refrain that dates to Mr. Trump’s first presidential campaign in 2016, when he often cited false statistics to claim historically high murder rates and record-breaking urban crime. After he was elected, those warnings waned, even though the country had its biggest one-year increase in murder in 2020, when he was in office.

Once he lost that election, though, Mr. Trump wasted no time in falsely claiming crime records, saying in a 2022 address that “our country is now a cesspool of crime.”

In making the case for a second term, Mr. Trump has stuck to that message, though his argument has evolved this election cycle from false claims on crime rates to an attack on the credibility of any evidence that refutes him. Here’s how.

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Ukrainian officials said on Monday that their forces had fired at North Korean soldiers in combat for the first time since their deployment by Russia to its western Kursk region.

The clashes mark the first direct intervention by a foreign army since Russia’s full-scale invasion of Ukraine began in 2022, as well as an expansion of what was already the largest land war in Europe since the second world war.

“The first military units of the DPRK [Democratic People’s Republic of Korea] have already come under fire in Kursk,” Andriy Kovalenko, Ukraine’s top counter-disinformation official within the national security council, said on Telegram. A senior Ukrainian intelligence official confirmed the military engagement to the Financial Times but declined to provide further details.

In Kyiv, foreign minister Andrii Sybiha said he had discussed with his visiting German counterpart Annalena Baerbock the “need for decisive action” in response to North Korea’s deepening involvement in the war.

“We urge Europe to realise that the DPRK troops are now carrying [out] an aggressive war in Europe against a sovereign European state,” Sybiha said in a news conference.

The US on Monday called out Russia and China at the UN Security Council for “shamelessly protecting” and emboldening North Korea. South Korea and the EU also condemned the deployment and expressed concern that Russia could reward North Korea with transfers of nuclear and ballistic technology.

Another senior Ukrainian official told the FT that Moscow was already providing military technologies to Pyongyang to help with its missile programmes, as well as “money”.

In Moscow, Russian President Vladimir Putin met North Korea’s foreign minister, Choe Son-hui, in the Kremlin on Monday.

Choe passed on a greeting from North Korean leader Kim Jong Un, who has backed Russia’s invasion of Ukraine and signed a treaty with Putin in June that includes a mutual security assistance clause.

The foreign minister last week said that North Korea had “no doubt whatsoever that under the wise leadership of the honourable Russian President Vladimir Putin, the Russian army and people will surely achieve a great victory in their sacred struggle to defend the sovereign rights and security of their state”.

Putin has not confirmed the North Korean deployment but he hinted at it last month, indicating it fell under the security provisions in the treaty.

US and South Korean officials last week confirmed Ukraine’s assessment that around 8,000 North Korean troops were sent to Kursk last month to help Russia’s army push Ukrainian forces out of territory they have occupied since August. Senior Ukrainian intelligence officials told the FT that the forces were in barracks about 50km from the Ukrainian border and preparing to enter the fight within “days”.

Kyiv, Washington and Seoul said that Pyongyang had sent roughly 12,000 troops in all to Russia for its ongoing war effort, including 500 officers and three generals. The remaining forces are located in Russia’s far east, where they are undergoing training.

The White House has said that the North Koreans would become “legitimate military targets” if they entered the fight against Ukraine.

Ukrainian President Volodymyr Zelenskyy said on Friday that Ukraine could see where Russia was gathering the North Koreans and urged western nations to lift restrictions on long-range weapons to “pre-emptively” strike them before they could attack his forces.

The senior Ukrainian intelligence official declined to provide specifics about the first military engagement between his country’s forces and the North Koreans. But he said that it occurred within Russia’s Kursk region, where Ukraine controls some 600 sq km of territory, or a little more than half of what it previously held following the summer incursion that took Moscow by surprise.

Ukraine’s military intelligence directorate, the GUR, said over the weekend that Russia had armed the North Korean troops in Kursk with 60mm mortars, assault rifles, machine guns, sniper rifles, anti-tank guided missiles and shoulder-launched anti-tank rocket launchers. The GUR said that some were also provided with night-vision devices and thermal imagers. A few hundred troops from North Korea’s special forces have also been deployed in Kursk.

Ukrainian officials and military analysts have raised questions about the quality and combat effectiveness of the North Korean troops, with most being described as inexperienced, low-ranking soldiers.

“We will know soon” how well they can fight, said one of the officials on Monday.