In 2023, our team resolved a 1985 conjecture posed by Paul Erdős. The work was the outcome of collaboration that combined geometry, graph theory, linear programming, harmonic analysis, and artificial intelligence. We have created a video presenting the main ideas of the proof visually, using animations, without requiring prior expertise in any of these areas. Watch on YouTube

When we ask large language models questions, they often seem to “know” facts about the world, such as which city is in which country, or what language is spoken where. But how do these models store and retrieve that knowledge? A recent study (Hernandez et al., 2023) found that relations like “capital of” are often represented by linear relationships in the latent space of models. Our research takes this line of work further by looking at collections of relations at once.

What fraction of the plane can be colored so that two colored points cannot be exactly a unit distance away from each other? This geometric question was formulated by Leo Moser in the early 1960s. According to a conjecture of Paul Erdős, this fraction must be less than ¼. The currently best lower bound of 0.2293 is given by a construction by Hallard Croft dating back to 1967. Several research groups have published partial results on the problem, gradually strengthening the initial upper density estimate of 0.