Abstract. Garbe et al. [ Ann. Inst. Henri Poincaré Probab. Stat., 60 (2024), pp. 2878–2922] recently introduced a general class of random graph processes called flip processes and proved that the typical evolution of these discrete-time random graph processes corresponds to certain continuous-time deterministic graphon trajectories. We obtain a complete characterization of the equivalence classes of flip process rules with the same graphon trajectories. As an application, we characterize the flip process rules which are unique in their equivalence classes. These include several natural families of rules, such as the complementing rules, the component completion rules, the extremist rules, and the clique removal rules.

We study minimum degree conditions under which a graph $G$ contains $k$th powers of paths and cycles of arbitrary specified lengths. We determine precise thresholds, assuming that the order of $G$ is large. This extends a result of Allen, Böttcher, and Hladký [ J. Lond. Math. Soc. (2), 84 (2011), pp. 269--302] concerning the containment of squares of paths and squares of cycles of arbitrary specified lengths and settles a conjecture of theirs in the affirmative.

We obtain an approximate sparse hypergraph version of the blow-up lemma, showing that partite hypergraphs with sufficient regularity of small subgraph counts behave as if they were complete partite for the purpose of embedding bounded degree hypergraphs.

Given integers $k,j$ with $1\le j \le k-1$, we consider the length of the longest $j$-tight path in the binomial random $k$-uniform hypergraph $H^k(n,p)$. We show that this length undergoes a phase transition from logarithmic length to linear and determine the critical threshold, as well as proving upper and lower bounds on the length in the subcritical and supercritical ranges. In particular, for the supercritical case we introduce the \tt Pathfinder algorithm, a depth-first search algorithm which discovers $j$-tight paths in a $k$-uniform hypergraph. We prove that, in the supercritical case, with high probability this algorithm will find a long $j$-tight path.