rainbow path; and the authors conjecture that there are Hamiltonian rainbow paths even if $k = a n$ for some suitably small constant factor $a$.

Paths with repeated colors
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Generalizing the notion of a rainbow path, we consider paths in $K_n$ that use every color at most a constant number of times. We call such paths *$k$-rainbow paths*, where $k$ is the number of times a color may appear on the path. We will prove that in every proper edge coloring of $K_n$ and for every integer $k>0$, there are $k$-rainbow paths on at least $n-O(n/k!)$ vertices. As far as we know, there are no previous results in this direction.

Note on Latin squares
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We now give some motivation for the study of rainbow cycles and paths by relating it to problems whose nature is not inherently graph-theoretic.

Latin squares have been a popular topic in combinatorics at least since the times of Euler, who studied them extensively. An array of $n$ rows and $n$ columns is called a *Latin square of order $n$* if every number in $[n] = \{1,\dotsc,n\}$ appears exactly once in each of its rows and columns. A (complete) *transversal* of a Latin square 