Definition: Two edges $\{x, y\}$ and $\{w, z\}$ of $G$ are said to be 3-disjoint if the induced subgraph of $G$ on $\{x, y, w, z\}$ consists of exactly two disjoint edges. (See page 5 of this file.)
I understand the definition meaning but i do not understand why do we call 3-disjoint, and what is the meaning of 3-disjoint ?
In the article they borrow the term from, the authors use $t$-disjoint to mean the edges are at distance at least $t$. Two edges in this particular graph $G$ that induce a subgraph of two disjoint edges are actually at distance at least $3$ (as opposed to $2$, as you might think), because the graph is bipartite.