The areas where the United States and China can work together seem to be shrinking fast, and the risks of confrontation are growing. But it was clear on Friday that both countries are trying to salvage what they can.

Preserving some semblance of cooperation — and the difficulty of doing so — was at the heart of a meeting between Secretary of State Antony J. Blinken and China’s leader, Xi Jinping, in Beijing on Friday. It was the latest effort by the rivals to keep communications open even as disputes escalate over trade, national security and geopolitical frictions.

Officials in both countries said they had made progress on a few smaller, pragmatic fronts, including setting up the first U.S.-China talks on artificial intelligence in the coming weeks. They also said they would continue improving communications between their militaries and increase cultural exchanges.

But on fundamental strategic issues, each side held little hope of moving the other, and they appeared wary of the possibility of sliding into further conflict.