Given integers k, j with 1 ≤ j ≤ 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 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.

Given integers k, j with 1 ≤ j ≤ 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 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.

We prove that certain classes of metrically homogeneous graphs omitting triangles of odd short perimeter as well as triangles of long perimeter have the extension property for partial automorphisms and we describe their Ramsey expansions.Dedicated to Norbert Sauer on the occasion of his 70th birthday.

No abstract

Fractional isomorphism is a well-studied relaxation of graph isomorphism with a very rich theory. Grebík and Rocha [Combinatorica 42, pp 365--404 (2022)] developed a concept of fractional isomorphism for graphons and proved that it enjoys an analogous theory. In particular, they proved that if $G_1,G_2,\ldots$ converge to a graphon $U$, $H_1,H_2,\ldots$ converge to a graphon $W$ and each $G_i$ is fractionally isomorphic to $H_i$, then $U$ is fractionally isomorphic to $W$. Answering the main question from \emph{ibid}, we prove the converse of the statement above: If $U$ and $W$ are fractionally isomorphic graphons, then there exist sequences of graphs $G_1,G_2,\ldots$ and $H_1,H_2,\ldots$ which converge to $U$ and $W$ respectively and for which each $G_i$ is fractionally isomorphic to $H_i$. As an easy but convenient corollary of our methods, we get that every regular graphon can be approximated by regular graphs.