THE DEAD are logged in a handwritten scrawl, each line a tragedy. “21 Park Street, two corpses, man and woman; Central Market, two corpses, one on the road, another in car; 101 Builders Avenue, one corpse, found under a blanket.” There are no names, no next of kin, just lines, and thousands of them. From March 12th the entries become infrequent and chaotic, not because there are fewer new deaths, but because counting them is impossible. The authorities say they have already certified 4,000 dead. A deputy mayor believes the current figure could be 20,000, or more. In truth, no one knows. But clearly it is already a very large number.

Mariupol, a city of 350,000, was surrounded by Russian forces in late February, and has since been subjected to a savage assault. The Russians targeted the essentials from the start: disabling the power supply, heating, water, communications and, later, the emergency services. From March 10th they began bombing from the air, hitting hospitals and bomb shelters among other civilian infrastructure. On March 16th, a particularly black day, Russian missiles fell on a swimming pool, a cinema, and a theatre known to be sheltering 1,300 people underground. The word “children”, written in large letters on the pavement outside, did nothing to prevent the attack. On March 20th another Russian bomb hit a school on the eastern side of the city, where 400 people were reportedly sheltering. Later that day, Russia issued an ultimatum to surrender Mariupol by 5am the following morning, which many interpreted as a warning of worse crimes to come. The Ukrainian government refused.

Eyewitnesses say a watershed was reached around March 15th, when Russian airstrikes intensified from a couple of bombs a day to more than 50. Each impact was terrifying, says Irina Perederey, a 30-year old council worker who was lucky to escape the city that day: “The shock waves made you convinced your house is about to collapse.” Oleksandr Horbachenko, an ambulance-service welder, says that things got progressively bleaker as the days wore on. By the time the 32-year-old left on March 18th, Mariupol was in a state of collapse, with no municipal services, no potable water, and starvation a growing threat for those trapped in the city. The last working supermarket closed its doors on around March 13th, after a missile fell through the roof, killing some of the customers. At least 80% of the city’s buildings are bombed out and uninhabitable, he says. “The whole of the centre is in ruins, with wires and glass everywhere. The worst thing is seeing the corpses strewn across the street. There are hundreds of them rotting away near the central market.”

The Economist understands that one hospital remains operational, despite an air strike that demolished its roof and top floors, killing patients and doctors in the process. The operating theatre, which was damaged in the blast, has been moved to the ground floor, where it is marginally more protected. But carpet-bombing across the city, not to mention a curfew, means it is hard to get the injured there. Even when they make it, it is difficult to find beds. One doctor said that diesel for the generator is a constant worry. The Ukrainian army has been providing it so far, but supplies are running low.

People want to leave, but getting out is risky. Talks have failed to produce a recognised humanitarian corridor out of Mariupol, such as now exists in around nine other places. Those who are mobile are taking matters into their own hands, leaving any way they can, often under artillery fire. But the fighting is intensifying, and that is making the already dangerous escape routes even more deadly. Part of the city is already in danger of falling. Andriy Biletsky, the founder of Azov regiment, a paramilitary outfit defending Mariupol alongside elite Ukrainian navy seals from the 36th Marine Brigade, confirmed that street-to-street fighting is under way in the east of the city. Elsewhere, defences are holding, he insisted. But the prospects do not look good: Ukrainian forces in Mariupol are vastly outnumbered, with 3,500 soldiers facing 14,000 invaders, around a tenth of the total estimated Russian force in the country.

The government in the capital appears to have ruled out the possibility of breaking the blockade by force. On March 19th Oleksiy Arestovych, a senior adviser to Ukraine's president, said the nearest available forces were more than 70 miles (110km) away and would have to traverse open terrain, completely exposed to Russian air attacks, to reach the city. “That leaves politics and diplomacy,” he said. Those in contact with people in Mariupol say that, however relief comes, it will have to come soon to be of any use. Serhiy Taruta, a local MP and the former governor of the Donetsk region, challenged the West to do more. He says that existing international pressure is nowhere near enough to stop Vladimir Putin’s “medieval” conduct. “Cut off Russian trade, finance, oil, and now, and perhaps he'll pause to think,” Mr Tartuta said. “Don’t do that, and you’ll have the blood of hundreds of thousands of Mariupol people on your hands.”