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Explore the properties of the unit distance graph of the plane in this hour-long conference talk by Máté Matolcsi at ICBS2024. Delve into the famous Hadwiger-Nelson problem, which seeks to determine the chromatic number of the unit distance graph UDG(ℝ²) of the plane. Discover the minimal number of colors needed to color the plane such that any pair of points at distance 1 have different colors. Examine related problems on various parameters of the unit distance graph, including a proof of Erdos' conjecture that any measurable subset of ℝ² avoiding the unit distance must have an upper density of
