A graph $G$ whose edges are coloured (not necessarily properly) contains a full rainbow matching if there is a matching $M$ that contains exactly one edge of each colour. We refute several conjectures on matchings in hypergraphs and full rainbow matc
hings in graphs, made by Aharoni and Berger and others.
In this note we show that for each Latin square $L$ of order $ngeq 2$, there exists a Latin square $L eq L$ of order $n$ such that $L$ and $L$ differ in at most $8sqrt{n}$ cells. Equivalently, each Latin square of order $n$ contains a Latin trade of
size at most $8sqrt{n}$. We also show that the size of the smallest defining set in a Latin square is $Omega(n^{3/2})$. %That is, there are constants $c$ and $n_0$ such that for any $n>n_0$ the size of the smallest defining %set of order $n$ is at least $cn^{3/2}$.
