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The celebrated conjecture of Lov\'asz from 1969 asks whether every connected vertex-transitive graph has a Hamiltonian path. Buci\'c, Christoph, Pokrovskiy and Steiner recently proved that every such graph on $n$ vertices contains a cycle of length $n^{2/3-o(1)}$. In this paper, we improve this bound to $n^{1-o(1)}$. O...
Let $C_{k}^{(r)}$ be the $r$-uniform linear cycle on $k$ hyperedges. An $r$-graph is $C_{k}^{(r)}$-free if it contains no copy of $C_{k}^{(r)}$. Let $\operatorname{ex}_{r}(n,C_{k}^{(r)})$ denote the maximum number of hyperedges in an $n$-vertex $C_{k}^{(r)}$-free $r$-graph. Balogh, Narayanan and Skokan asked whether, f...
Let $C_{k}^{(r)}$ be the $r$-uniform linear cycle on $k$ hyperedges. An $r$-graph is $C_{k}^{(r)}$-free if it contains no copy of $C_{k}^{(r)}$. Let $\operatorname{ex}_{r}(n,C_{k}^{(r)})$ denote the maximum number of hyperedges in an $n$-vertex $C_{k}^{(r)}$-free $r$-graph. Balogh, Narayanan and Skokan asked whether, f...
Given any fixed integer $k \ge 2$ and sufficiently large $d$, we show that the largest possible fractional chromatic number of a $k$-uniform $d$-degenerate uncrowded hypergraph $H$ (i.e., with girth at least $5$) satisfies \[ \chi_f(H) = (1 + o_d(1)) \left((k-1)\,\frac{d}{\log d}\right)^{\frac{1}{k-1}}. \] In fact, we...
The celebrated conjecture of Lov\'asz from 1969 asks whether every connected vertex-transitive graph has a Hamiltonian path. Buci\'c, Christoph, Pokrovskiy and Steiner recently proved that every such graph on $n$ vertices contains a cycle of length $n^{2/3-o(1)}$. In this paper, we improve this bound to $n^{1-o(1)}$. O...
The search for Hamiltonian cycles in vertex-transitive graphs and digraphs is a classical problem at the interface of graph theory and group theory. In the undirected setting, this goes back to the well-known conjectures of Lov\'asz and Thomassen concerning Hamiltonian paths and cycles in connected vertex-transitive gr...
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