  We study edge-labelings of the complete bidirected graph
$\overset{\tiny\leftrightarrow}{K}_n$ with functions from the set $[d] = \{1,
\dots, d\}$ to itself. We call a cycle in $\overset{\tiny\leftrightarrow}{K}_n$
a fixed-point cycle if composing the labels of its edges results in a map that
has a fixed point, and we say that a labeling is fixed-point-free if no
fixed-point cycle exists. For a given $d$, we ask for the largest value of $n$,
denoted $R_f(d)$, for which there exists a fixed-point-free labeling of
$\overset{\tiny\leftrightarrow}{K}_n$. Determining $R_f(d)$ for all $d >0$ is a
natural Ramsey-type question, generalizing some well-studied zero-sum problems
in extremal combinatorics. The problem was recently introduced by Chaudhury,
Garg, Mehlhorn, Mehta, and Misra, who proved that $d \leq R_f(d) \leq d^4+d$
and showed that the problem has close connections to EFX allocations, a central
problem of fair allocation in social choice theory.
  In this paper we show the improved bound $R_f(d) \leq d^{2 + o(1)}$, yielding
an efficient ${{(1-\varepsilon)}}$-EFX allocation with $n$ agents and
$O(n^{0.67})$ unallocated goods for any constant $\varepsilon \in (0,1/2]$;
this improves the bound of $O(n^{0.8})$ of Chaudhury, Garg, Mehlhorn, Mehta,
and Misra.
  Additionally, we prove the stronger upper bound $2d-2$, in the case where all
edge-labels are permulations. A very special case of this problem, that of
finding zero-sum cycles in digraphs whose edges are labeled with elements of
$\mathbb{Z}_d$, was recently considered by Alon and Krivelevich and by
M\'{e}sz\'{a}ros and Steiner. Our result improves the bounds obtained by these
authors and extends them to labelings from an arbitrary (not necessarily
commutative) group, while also simplifying the proof.
