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  "body": "\n## Statement\n\nIn 1962, Lajos Pósa conjectured that every $n$-vertex graph with minimum degree at least $\\frac{2n}3$ contains the square of a Hamiltonian cycle, generalizing Dirac's theorem. The conjecture was further extended by Paul Seymour in 1974 to the following:\n\nEvery graph on $n$ vertices with minimum degree at least $\\frac{kn}{k+1}$ contains the $k$-th power of a Hamiltonian cycle.\n\nThe blow-up lemma was used by Komlós, Sárközy, and Szemerédi to prove the conjecture for all sufficiently large values of $n$ (for a fixed $k$) in 1998.\n",
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