First off I want to thank anyone who takes the time to read this and I apologize if I am bad at explaining this I am not the best at math, but I have been trying really hard to search through the internet and even asking others with no success.
Example:[A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z, F, F, N, N, A, A, A, !, !, !, T, T, T, T, T, T]
[A, B] would be the considered the same as [B, A]. [T, T] would not be valid either as objects shouldn't be paired with them selves (think of each object as a person looking for a partner but they can only be partnered with each other once)
Real World Example:
Let's say you have a sign up sheet every one can sign their names on this sheet and they can sign their names multiple times if they want to, every person who signs their name gets paired with some one and if you signed your name multiple times you can't pair with the same person more then once.
Adam,Bob,Charlie,Dan,Erica,Fred,Gina,Hailey,Ivan,Jessica,Kathy,Laura,Marcus,Nate,Ocho,Pierce,Quinn,Rich,Sally,Tom,Uruka,Valerie,Wilson,XQC,Yura,Zach,Adam,Adam,Adam,Bob,Bob,Bob,Charlie,Charlie,Charlie,Gina,Gina,Gina,Marcus,Marcus,Marcus
Marcus has his name multiple times Same with Adam but Adam and Marcus can only be partners 1 time, every other time they need to find some one different..
I am basically trying to find out how can I detect the breaking point where there are too many duplicate sign ups which would exhaust the amount of unique pairs. Because at some point you can only match people so many times before they cannot match with any one new any more or they are the only ones left in the list of names.
Order your $n$ people by the number of times they sign up, from largest to smallest, and call these numbers $d_1 \ge d_2 \ge \cdots \ge d_n$. Treat each person as the vertex of a graph. Pairing the people is analogous to drawing an edge between their vertices, and the assumption that a person cannot pair with him/herself, nor can people pair more than once, ensures that the resulting graph is simple, and that $(d_1,\ldots,d_n)$ is the degree sequence of the graph. Your existence question is then equivalent to the question of whether the degree sequence above can be realized in a simple graph, which can be determined for concrete examples using the Erdős-Gallai Theorem.