This paper studies the fundamental relations among integer flows, modulo orientations, integer-valued and real-valued circular flows, and monotonicity of flows in signed graphs. A (signed) graph is modulo-$(2p+1)$-orientable if it has an orientation such that the indegree is congruent to the outdegree modulo $2p+1$ at each vertex. An integer-valued $\frac{2p+1}{p}$-flow is a flow taking integer values in $\{\pm p, \pm (p+1)\}$. Extending a fundamental result of Jaeger to signed graphs, we show that a bridgeless signed graph is modulo-$(2p+1)$-orientable if and only if it admits an integer-valued $\frac{2p+1}{p}$-flow. It was conjectured by Raspaud and Zhu that, for any signed graph, the admission of a circular $r$-flow implies the admission of an integer-valued $\lceil r \rceil$-flow. Although this conjecture has been disproved in general, it is confirmed in this paper for bridgeless signed graphs if $r=\frac{2p+1}{p}$ and $p \geq 3$.

The anti-Ramsey problem was introduced by Erdös, Simonovits, and Sós in 1970s. The anti-Ramsey number of a hypergraph H, ar(n,s, H), is the smallest integer c such that in any coloring of the edges of the s-uniform complete hypergraph on n vertices with exactly c colors, there is a copy of H whose edges have distinct colors. In this paper, we determine the anti-Ramsey numbers of linear paths and loose paths in hypergraphs for sufficiently large n and give bounds for the anti-Ramsey numbers of Berge paths. Similar exact anti-Ramsey numbers are obtained for linear/loose cycles, and bounds are obtained for Berge cycles. Our main tools are the path extension technique and stability results on hypergraph Turán problems of paths and cycles.