Rainbow Erdős matching conjecture

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. Rainbow Erdős Matching Conjecture

   Let $\binom{[n]}{k}$ denote the family of $k$-subsets of $[n]$. Rainbow Erdős Matching Conjecture. If $n\geq(s+1)k$ and $\mathcal{F}_1,\dots,\mathcal{F}_{s+1}\subseteq\binom{[n]}{k}$ have no pairwise disjoint $F_1,\dots,F_{s+1}$ with $F_i\in\mathcal{F}_i$ for every $i$, then

   $$\min_{i\in[s+1]}|\mathcal{F}_i|\leq\max\left\{\binom{(s+1)k-1}{k},\binom{n}{k}-\binom{n-s}{k}\right\}.$$

   This is the multipartite, or rainbow, extension of the Erdős Matching Conjecture; the source gives no resolution status.

   source: Jiuqiang Liu, Guihai Yu, Lihua Feng and Yongtao Li, “L-intersecting or Configuration Forbidden Families on Set Systems and Vector Spaces over Finite Fields”, arXiv:2403.04289 (2024).