62nd Ukrainian National Mathematical Olympiad

Number the vertices of the graph with numbers from 1 to 4046. Consider the following graph: all edges between vertices of different parity are colored 1 and all other edges are colored in their unique colors. Consider any partition of the vertices into 2023 into pairs. Suppose that in $x$ of these pairs the vertices have different parity, and in $(2023-x)$ the same.

Note that $(2023-x)$ is an even number, because even numbers that are not in $x$ pairs where the numbers are of different parity must be paired with each other. Thus, $x$ is odd, and we have at least one edge of color 1 we have. Thus, the total number of different colors is $$1_{\{\text{first color}\}}+(2023-x{)}_{\{\text{different colors in pairs with same parity}\}}=2024-x,$$ is an odd number, i.e. we cannot get exactly 1000 colors.