Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5777648 | Journal of Combinatorial Theory, Series B | 2017 | 12 Pages |
Abstract
Motivated by a question of Grinblat, we study the minimal number v(n) that satisfies the following. If A1,â¦,An are equivalence relations on a set X such that for every iâ[n] there are at least v(n) elements whose equivalence classes with respect to Ai are nontrivial, then A1,â¦,An contain a rainbow matching, i.e. there exist 2n distinct elements x1,y1,â¦,xn,ynâX with xiâ¼Aiyi for each iâ[n]. Grinblat asked whether v(n)=3nâ2 for every nâ¥4. The best-known upper bound was v(n)â¤16n/5+O(1) due to Nivasch and Omri. Transferring the problem into the setting of edge-coloured multigraphs, we affirm Grinblat's question asymptotically, i.e. we show that v(n)=3n+o(n).
Related Topics
Physical Sciences and Engineering
Mathematics
Discrete Mathematics and Combinatorics
Authors
Dennis Clemens, Julia Ehrenmüller, Alexey Pokrovskiy,