Random cliques in random graphs

We show that for each $r\ge 4$, in a density range extending up to, and slightly beyond, the threshold for a $K_r$-factor, the copies of $K_r$ in the random graph $G(n,p)$ are randomly distributed, in the (one-sided) sense that the hypergraph that they form contains a copy of a binomial random hypergraph with almost exactly the right density. Thus, an asymptotically sharp bound for the threshold in Shamir's hypergraph matching problem -- recently announced by Jeff Kahn -- implies a corresponding bound for the threshold for $G(n,p)$ to contain a $K_r$-factor. We also prove a slightly weaker result for $r=3$, and (weaker) generalizations replacing $K_r$ by certain other graphs $F$. As an application of the latter we find, up to a log factor, the threshold for $G(n,p)$ to contain an $F$-factor when $F$ is $1$-balanced but not strictly $1$-balanced.Comment: 19 pages; expanded introduction and Section 5, plus minor correction

Threshold for Steiner triple systems

We prove that with high probability $\mathbb{G}^{(3)}(n,n^{-1+o(1)})$ contains a spanning Steiner triple system for $n\equiv 1,3\pmod{6}$, establishing the tight exponent for the threshold probability for existence of a Steiner triple system. We also prove the analogous theorem for Latin squares. Our result follows from a novel bootstrapping scheme that utilizes iterative absorption as well as the connection between thresholds and fractional expectation-thresholds established by Frankston, Kahn, Narayanan, and Park.Comment: 22 pages, 1 figur

Perfect matchings in random sparsifications of Dirac hypergraphs

For all integers $n \geq k > d \geq 1$, let $m_{d}(k,n)$ be the minimum integer $D \geq 0$ such that every $k$-uniform $n$-vertex hypergraph $\mathcal H$ with minimum $d$-degree $\delta_{d}(\mathcal H)$ at least $D$ has an optimal matching. For every fixed integer $k \geq 3$, we show that for $n \in k \mathbb{N}$ and $p = \Omega(n^{-k+1} \log n)$, if $\mathcal H$ is an $n$-vertex $k$-uniform hypergraph with $\delta_{k-1}(\mathcal H) \geq m_{k-1}(k,n)$, then a.a.s.\ its $p$-random subhypergraph $\mathcal H_p$ contains a perfect matching ($m_{k-1}(k,n)$ was determined by R\"{o}dl, Ruci\'nski, and Szemer\'edi for all large $n \in k\mathbb N$). Moreover, for every fixed integer $d < k$ and $\gamma > 0$, we show that the same conclusion holds if $\mathcal H$ is an $n$-vertex $k$-uniform hypergraph with $\delta_d(\mathcal H) \geq m_{d}(k,n) + \gamma\binom{n - d}{k - d}$. Both of these results strengthen Johansson, Kahn, and Vu's seminal solution to Shamir's problem and can be viewed as "robust" versions of hypergraph Dirac-type results. In addition, we also show that in both cases above, $\mathcal H$ has at least $\exp((1-1/k)n \log n - \Theta (n))$ many perfect matchings, which is best possible up to a $\exp(\Theta(n))$ factor.Comment: 25 pages + 2 page appendix; Theorem 1.5 was proved in independent work of Pham, Sah, Sawhney, and Simkin (arxiv:2210.03064