Fixed-point cycles and EFX allocations

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 Mészáros 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.