arXiv subjects
Publications and source records attributed to Michael Simkin.
Let $H$ be a random $k$-uniform $n$-vertex hypergraph where every $k$-tuple belongs to $H$ independently with probability $p$. We show that for some $\varepsilon_k > 0$, if $p \geq n^{-\varepsilon_k}$, then asymptotically almost surely $H$ contains an $\left( n , k , k - 1 \right)$-Steiner System. Our main tool is Keevash's method of Randomized Algebraic Constructions.
This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erdős-Szekeres theorem: For every $k\ge1$, every order-$n$ $k$-dimensional permutation contains a monotone subsequence of length $Ω_{k}\left(\sqrt{n}\right)$, and this is tight. On the other hand, and unlike the classical case, the longest monotone subsequence in a random $k$-dimensional permutation of order $n$ is asymptotically almost surely $Θ_{k}\left(n^{\frac{k}{k+1}}\right)$.