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We prove a lower bound on the length of the longest $j$-tight cycle in a $k$-uniform binomial random hypergraph for any $2 le j le k-1$. We first prove the existence of a $j$-tight path of the required length. The standard sprinkling argument is not enough to show that this path can be closed to a $j$-tight cycle -- we therefore show that the path has many extensions, which is sufficient to allow the sprinkling to close the cycle.
Given integers $k,j$ with $1le j le k-1$, 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 deter
In 1999, Katona and Kierstead conjectured that if a $k$-uniform hypergraph $cal H$ on $n$ vertices has minimum co-degree $lfloor frac{n-k+3}{2}rfloor$, i.e., each set of $k-1$ vertices is contained in at least $lfloor frac{n-k+3}{2}rfloor$ edges, the
Inspired by the study of loose cycles in hypergraphs, we define the emph{loose core} in hypergraphs as a structure which mirrors the close relationship between cycles and $2$-cores in graphs. We prove that in the $r$-uniform binomial random hypergrap
In an $r$-uniform hypergraph on $n$ vertices a tight Hamilton cycle consists of $n$ edges such that there exists a cyclic ordering of the vertices where the edges correspond to consecutive segments of $r$ vertices. We provide a first deterministic po
We show that, for a natural notion of quasirandomness in $k$-uniform hypergraphs, any quasirandom $k$-uniform hypergraph on $n$ vertices with constant edge density and minimum vertex degree $Omega(n^{k-1})$ contains a loose Hamilton cycle. We also gi