Discrepancies of perfect matchings in hypergraphs

Abstract

In this paper, we determine the minimum degree threshold of perfect matchings with high discrepancy in $r$-edge-colored $k$-uniform hypergraphs for all $k\ge 3$ and $r\ge 2$, thereby completing the investigation into discrepancies of perfect matchings that has recently attracted significant attention. Our approach identifies this discrepancy threshold with a novel family of multicolored uniform hypergraphs and reveals new phenomena not covered in previous studies. In particular, our results address a question of Balogh, Treglown and Zárate-Guerén concerning 3-uniform hypergraphs.

25 pages with 1 figure; added acknowledgements and a new appendix to address a missing case in the original Lemma 12