In Sikinia, there are $2024$ cities. Between some of them there are flight connections, which can be used in either direction. No city has a direct flight to all $2023$ other cities. It is known, however, that there is a positive integer $n$ with the following property: For any $n$ cities in Sikinia, there is another city which is directly connected to all these cities. Determine the largest possible value of $n$.
Problem
Source: Bundeswettbewerb Mathematik 2024, Round 2 - Problem 4
Tags: combinatorics, combinatorics proposed, graph theory