UA-UMO 2023 Problem 108

Consider a complete graph on $4046$ nodes, whose edges are colored in some colors. Let's call this graph $k$ -good if we can split all its nodes into $2023$ pairs so that there are exactly $k$ distinct colors among the colors of $2023$ edges that connect the nodes from the same pairs. Is it possible that the graph is $999$ -good and $1001$ -good but not $1000$ -good?

Proposed by Anton Trygub