Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Source: original paper, printed p. 531 and reference [2] on p. 557.
The triangular-lattice proof of Theorem 1 imports the following exact statement: every two-coloring of contains a monochromatic congruent copy of , three collinear points with adjacent distance one. The source cites Theorem 8 of Erdős–Graham–Montgomery–Rothschild–Spencer–Straus, Euclidean Ramsey Theorems, I, Journal of Combinatorial Theory, Series A 14 (1973), 341–363. The full earlier proof is linked at Paper I, Theorem 8, including its equilateral-triangle prerequisite. It is not duplicated here.
The stronger planar Theorem 1′ has a separate elementary two-circle proof and does not use this input. The source's two methods are retained separately.
The finite constructions also use elementary Euclidean distance calculations, finite pigeonhole counting and basic facts about simple graphs. Their actual deductions are supplied in the corresponding proof pages. The later infinite-configuration and edge-coloring sections have different topological, set-theoretic and Ramsey inputs; they are not covered by this finite-source proof claim.