On Erdős-Rado numbers (Q1842571)

The Erdős-Rado number \(\text{ER}(k, l)\) is the smallest \(n\) such that if the \(k\)-subsets of \(\{1,\dots, n\}\) are colored with an arbitrary number of colors, then there is an \(l\)-subset of \(\{1,\dots, n\}\) whose \(k\)-subsets are colored canonically. The existence of such a number is guaranteed by the canonical Ramsey theorem. The authors prove new bounds for the Erdős-Rado numbers, the most important is \(2^{c_ 1 l^ 2}\leq \text{ER}(2, l)\leq 2^{c_ 2 l^ 2\log l}\).