Perfect matchings in random sparsifications of Dirac hypergraphs

For $d\in\mathbb{N}$, the minimum $d$-degree $\delta_{d}(\mathcal{H})$ of a hypergraph $\mathcal{H}$ is the minimum of $|\{e\in\mathcal{H}:v_{1},\dots,v_{d}\in e\}|$ among all choices of $d$ distinct vertices $v_{1},\dots,v_{d}\in V(\mathcal{H})$. If $\mathcal{H}$ is $k$-uniform, we also call $\delta_{k-1}(\mathcal{H})$ the minimum codegree. For $n\geq k>d\geq 1$, let $m_{d}(k,n)$ be the minimum integer $D\geq 0$ such that every $k$-uniform $n$-vertex hypergraph $\mathcal{H}$ with $\delta_{d}(\mathcal{H})\geq D$ has an optimal matching, and for each $s\in\{0,\dots,k-1\}$, let

Determining the value of $m_{d}(k,n)$, or even just $\overline{\mu_{d}}^{(s)}(k)$ in many cases, is a major open problem. Rödl, Ruciński, and Szemerédi [65] first proved that $m_{k-1}(k,n)\leq n/2+o(n)$ for $n\in k\mathbb{N}$ (in fact, they showed a tight Hamilton cycle exists if this codegree condition holds). This bound was improved by Kühn and Osthus [52] to $n/2+3k^{2}\sqrt{n\log n}$, and Rödl, Ruciński, and Szemerédi [62] improved it further to $n/2+O(\log n)$. Finally, Rödl, Ruciński, and Szemerédi [63] determined $m_{k-1}(k,n)=n/2-k+C(k,n)$ for all sufficiently large $n\in k\mathbb{N}$, with $C(k,n)\in\{3/2,2,5/2,3\}$ depending on $k$ and $n$. Rödl, Ruciński, Schacht, and Szemerédi [64] also gave a simple proof for a bound of $n/2+k/4$, that does not require $n$ to be large.

For $1\leq d\leq 3k/8$ and $n\in k\mathbb{N}$, both the exact and the asymptotic values of $m_{d}(k,n)$ are unknown for many cases. The exact value of $m_{d}(k,n)$ is known for $d\geq 3k/8$ and large $n\in k\mathbb{N}$ by a combination of results [21, 71], where $m_{d}(k,n)=(1/2+o(1))\binom{n-d}{k-d}$ and the exact bound of $m_{d}(k,n)$ follows from the obstructions called divisibility barriers. Khan [45] and independently Kühn, Osthus, and Treglown [55] showed that $m_{1}(3,n)=\binom{n-1}{2}-\binom{2n/3}{2}$ for large $n\in 3\mathbb{N}$. Khan [46] showed that $m_{1}(4,n)=\binom{n-1}{3}-\binom{3n/4}{3}$ for large $n\in 4\mathbb{N}$. Alon, Frankl, Huang, Rödl, Ruciński, and Sudakov [2] related the asymptotics of $m_{d}(k,n)$ and $m_{d}^{*}(k,n)$, where $m_{d}^{*}(k,n)$ is the minimum $D$ such that every $n$-vertex $k$-uniform hypergraph with minimum $d$-degree at least $D$ has fractional matching number $n/k$. Ferber and Kwan [19] showed that the limit of $m_{d}(k,n)/\binom{n-d}{k-d}$ exists as $n\in k\mathbb{N}$ tends to infinity, and it is conjectured [34, 53] that $\overline{\mu_{d}}^{(0)}(k)=\max\{1/2,1-(\frac{k-1}{k})^{k-d}\}$. See [72, Conjecture 1.5] for the exact conjectured value of $m_{d}(k,n)$ when $n\in k\mathbb{N}$ is large.

For the other case $k\nmid n$, Rödl, Ruciński, and Szemerédi [63] showed that $m_{k-1}(k,n)\leq n/k+O(\log n)$, and Han [35] determined $m_{k-1}(k,n)=\lfloor n/k\rfloor$ for all sufficiently large $n$ (and thus $\overline{\mu_{k-1}}^{(s)}(k)=1/k$ for all $s\neq 0$). Han [36, Conjecture 1.10] conjectured an upper bound on $\overline{\mu_{d}}^{(s)}(k)$ for all $d,s\in[k-1]$ and proved a matching lower bound on $m_{d}(k,n)$. (Thus, if true, Han’s conjecture implies the limit of $m_{d}(k,n)/\binom{n-d}{k-d}$ exists as $n\in k\mathbb{N}+s$ tends to infinity.) See [8, 36, 57] for more background on the non-divisible case, and for more discussion of this topic, see the surveys [61, 72].

A random $k$-uniform $n$-vertex hypergraph $\mathcal{H}^{k}(n,p(n))$ is a $k$-uniform hypergraph on $n$ vertices obtained by choosing each subset of $k$ vertices to be an edge with probability $p(n)$ independently at random. Regarding the existence of a perfect matching in a random hypergraph, it is natural to ask for the threshold for $\mathcal{H}^{k}(kn,p(n))$ to contain a perfect matching.

For $k=2$, in a seminal paper Erdős and Rényi [14] determined the (sharp) threshold for $\mathcal{H}^{2}(2n,p(n))$ to contain a perfect matching. They showed that the probability that $\mathcal{H}^{k}(2n,p(n))$ has a perfect matching tends to 1 if $p(n)=\frac{\log n+\omega(1)}{2n}$ and tends to 0 if $p(n)=\frac{\log n-\omega(1)}{2n}$.

On the other hand, for $k\geq 3$, it is much more difficult to determine the threshold for the appearance of a perfect matching. In 1979, Shamir (see [15, 67]) asked for the threshold for $\mathcal{H}^{k}(kn,p(n))$ to contain a perfect matching (a precise and explicit statement was mentioned in [11]). Schmidt and Shamir [67] showed that asymptotically almost surely (which we abbreviate as a.a.s.) $\mathcal{H}^{k}(kn,p(n))$ has a perfect matching if $p(n)=\omega(n^{-k+3/2})$. This was further improved by Frieze and Janson [23] to $p(n)=\omega(n^{-k+4/3})$. Finally, Johansson, Kahn, and Vu [37] proved that the threshold for $\mathcal{H}^{k}(kn,p(n))$ to contain a perfect matching is $\Theta(n^{-k+1}\log n)$, matching the threshold for $\mathcal{H}^{k}(kn,p(n))$ not to contain an isolated vertex. Recently, Kahn [40] determined the sharp threshold for $\mathcal{H}^{k}(kn,p(n))$ to contain a perfect matching, as well as the hitting time result [39], which proves the conjecture in [11] in a stronger form.

For any hypergraph $\mathcal{H}$ and $p\in[0,1]$, let $\mathcal{H}_{p}$ be a spanning random subhypergraph of $\mathcal{H}$ obtained by choosing each edge $e\in\mathcal{H}$ with probability $p$ independently at random. The problem of determining whether a certain property of the original hypergraph $\mathcal{H}$ is retained by $\mathcal{H}_{p}$ has been studied extensively [1, 3, 10, 28, 38, 50, 51, 59], and results of this nature are referred to as robustness results [69]. For example, Krivelevich, Lee, and Sudakov [50] showed a robust version of Dirac’s theorem that for every $n$-vertex graph $G$ with minimum degree at least $n/2$, a.a.s. its random subgraph $G_{p}$ contains a Hamilton cycle for $p=p(n)\geq C\log n/n$ for some absolute constant $C>0$, providing a common generalization of Dirac’s theorem [12] (when $p=1$) and the classic result of Pósa [60] (when $G\cong K_{n})$ on the threshold for the appearance of a Hamilton cycle in a random graph.

Our first result is the following robust version of hypergraph Dirac-type results on $m_{d}(k,n)$ in the general case $1\leq d\leq k-1$. Here, the integer $n$ is not necessarily divisible by $k$. The case $k=2$ follows from the result by Krivelevich, Lee, and Sudakov [50] on the robust Hamiltonicity.

Combining this result with the aforementioned prior work determining $m_{d}(k,n)$ [2, 21, 63, 71], we simultaneously obtain that for every $\gamma>0$, as $n\rightarrow\infty$, for $p=\Omega(\log n/n^{k-1})$, $\mathcal{H}_{p}$ a.a.s. has a perfect matching when $\mathcal{H}$ is a $k$-uniform $n$-vertex hypergraph satisfying $\delta_{d}(\mathcal{H})\geq(1/2+\gamma)\binom{n-d}{k-d}$ for some $d\geq 3k/8$ when $k\mid n$ and that $\mathcal{H}_{p}$ a.a.s. has an optimal matching when $\mathcal{H}$ is a $k$-uniform $n$-vertex hypergraph satisfying $\delta_{k-1}(\mathcal{H})\geq(1/k+\gamma)n$ when $k\nmid n$. Another interesting feature of this result is that it implies the existence of optimal matchings in random sparsifications of hypergraphs with minimum $d$-degree at least $\left(\overline{\mu_{d}}^{(s)}(k)+\gamma\right)\binom{n-d}{k-d}$ even in the cases in which the value of $\overline{\mu_{d}}^{(s)}(k)$ is not known. Since $\lim_{n\rightarrow\infty:k\mid n}\left.m_{d}(k,n)\middle/\binom{n-d}{k-d}\right.=\overline{\mu_{d}}^{(0)}(k)$ [19], for $n\in k\mathbb{N}$, the minimum degree condition in Theorem 1.1 can be replaced by $\delta_{d}(\mathcal{H})\geq m_{d}(k,n)+\gamma\binom{n-d}{k-d}$.

Our main result is the following robust version of the Dirac-type result by Rödl, Ruciński, and Szemerédi [63].

The value of $p$ in both Theorems 1.1 and 1.2 is asymptotically best possible, since it is well known that a.a.s. there are $\omega(1)$ isolated vertices in a random $k$-uniform $n$-vertex hypergraph $\mathcal{H}^{k}(n,p)$ if $p\leq\frac{(k-1)!\log n-\omega(1)}{n^{k-1}}$. In fact, both results generalize Johansson, Kahn, and Vu’s [37] solution to Shamir’s problem that the threshold for the existence of a perfect matching in $\mathcal{H}^{k}(kn,p(n))$ is $\Theta(n^{-k+1}\log n)$.

We remark that Theorems 1.1 and 1.2 are implied by Theorems 1.5 and 1.6 below, respectively.

To prove Theorems 1.1 and 1.2, we use the fractional version of the Kahn–Kalai conjecture [41] (conjectured by Talagrand [70]), recently resolved by Frankston, Kahn, Narayanan, and Park [22]. The Kahn–Kalai conjecture was recently proved in full by Park and Pham [58], but the fractional version is sufficient for our application. A precursor to these results was the main technical ingredient in Alweiss, Lovett, Wu, and Zhang’s [4] breakthrough on the Erdős–Rado sunflower conjecture [16], and the results have been found to have many additional applications. This paper, and the independent work of Pham, Sah, Sawhney, and Simkin [59] (discussed further in the remark at the end of this subsection), are the first to demonstrate an application of the result to robustness of Dirac-type results.

The Frankston–Kahn–Narayanan–Park theorem implies the Johansson–Kahn–Vu solution to Shamir’s problem. Moreover, it reduces our problem to proving that there exists a probability measure on the set of perfect or optimal matchings that is ‘well-spread’. Roughly speaking, this means that the probability measure chooses a perfect matching at random in such a way that no particular set of edges is very likely to be contained in the matching.

The next theorem follows from [22, Theorem 1.6]. More precisely, it follows from the derivation of [22, Theorem 1.1] from [22, Theorem 1.6].

In particular, by Theorem 1.4, it suffices to prove the following results to deduce Theorems 1.1 and 1.2, respectively.

For a $k$-uniform $n$-vertex hypergraph $\mathcal{H}$ with $\delta_{k-1}(\mathcal{H})\geq\delta n$ for some $\delta>1/2$, there are some earlier results [17, 18, 29] on counting the number of perfect matchings in $\mathcal{H}$. Recently, Glock, Gould, Joos, Kühn, and Osthus [29] showed that $\mathcal{H}$ has at least $\exp((1-1/k)n\log n-\Theta(n))$ perfect matchings, which is best possible up to an $\exp(\Theta(n))$ factor (this is also implicit in [18]), since this is also an upper bound for the number of perfect matchings in an $n$-vertex $k$-uniform complete hypergraph. Very recently, Ferber, Hardiman, and Mond [17] sharpened the bound further by showing that $\mathcal{H}$ has at least $(1-o(1))^{n}|\mathcal{M}(\mathcal{K}_{n}^{k})|\delta^{n/k}$ perfect matchings, where $|\mathcal{M}(\mathcal{K}_{n}^{k})|$ is the number of perfect matchings in an $n$-vertex complete $k$-uniform hypergraph.

As a corollary to Theorems 1.5 and 1.6, we extend the above results of [18, 29] to $n$-vertex $k$-uniform hypergraphs with minimum $d$-degree at least $m_{d}(k,n)+o(n^{k-d})$ or minimum codegree at least $m_{k-1}(k,n)$. Our bounds are best possible up to an $\exp(\Theta(n))$ factor.

For $k\in\mathbb{N}$, we let $[k]\coloneqq\{1,\dots,k\}$. For a $k$-uniform hypergraph $\mathcal{H}$ and an edge $e\in\mathcal{H}$, we often denote by $V(e)$ the set of vertices incident to $e$. For any set $S\subseteq V(\mathcal{H})$, we denote by $\mathcal{H}[S]$ the subgraph of $\mathcal{H}$ induced by $S$. For sets $S\subseteq V(\mathcal{H})$ and $\mathcal{S}\subseteq\binom{V(\mathcal{H})}{k-\left\lvert S\right\rvert}$, we denote by $N_{\mathcal{H}}(S;\mathcal{S})$ the set of edges $e\in\mathcal{H}$ such that $e=S\cup S^{\prime}$ for some $S^{\prime}\in\mathcal{S}$, and $d_{\mathcal{H}}(S;\mathcal{S})\coloneqq|N_{\mathcal{H}}(S;\mathcal{S})|$. We often omit $\mathcal{S}$ if $\mathcal{S}=\binom{V(\mathcal{H})}{k-\left\lvert S\right\rvert}$ (for example, we write $N_{\mathcal{H}}(S)$ and $d_{\mathcal{H}}(S)$). If $\left\lvert S\right\rvert=k-1$ and $U\subseteq V(\mathcal{H})$, we abuse notation and write $d_{\mathcal{H}}(S;U)$ for $d_{\mathcal{H}}(S;\{\{u\}\colon u\in U\})$. Moreover, for $v\in V(\mathcal{H})$, we write $d_{\mathcal{H}}(v;\mathcal{S})$ for $d_{\mathcal{H}}(\{v\};\mathcal{S})$. For disjoint subsets $W_{1},\dots,W_{k}\subseteq V(\mathcal{H})$, we denote by $e_{\mathcal{H}}(W_{1},\dots,W_{k})$ the number of edges $e\in\mathcal{H}$ with $|e\cap W_{i}|=1$ for all $i\in[k]$. We denote by $\overline{\mathcal{H}}$ the complement of a $k$-uniform hypergraph $\mathcal{H}$ such that $V(\overline{\mathcal{H}})\coloneqq V(\mathcal{H})$ and $\overline{\mathcal{H}}\coloneqq\binom{V(\mathcal{H})}{k}\setminus\mathcal{H}$. We take all asymptotic notations $o(\cdot),O(\cdot),\Theta(\cdot),\omega(\cdot),\Omega(\cdot)$ to be as $n\to\infty$, and all of their leading coefficients may depend on parameters other than $n$. We say that an event $\mathcal{E}$ holds asymptotically almost surely (a.a.s.) if $\mathbb{P}\left[\mathcal{E}\right]=1-o(1)$ as $n\rightarrow\infty$. For real numbers $x$, $y$, $\alpha$, and $\beta$ with $\beta\geq 0$, we write $x=(\alpha\pm\beta)y$ for $(\alpha-\beta)y\leq x\leq(\alpha+\beta)y$. We sometimes state a result with a hierarchy of constants which are chosen from right to left. If we state that the result holds whenever $a\ll b_{1},\dots,b_{t}$, then this means that there exists a function $f\colon(0,1)^{t}\to(0,1)$ such that $f(b_{1},\dots,\widetilde{b_{i}},\dots,b_{t})\leq f(b_{1},\dots,b_{i},\dots,b_{t})$ for $0<\widetilde{b_{i}}\leq b_{i}<1$ for all $i\in[t]$ and the result holds for all real numbers $0<a,b_{1},\dots,b_{t}<1$ with $a\leq f(b_{1},\dots,b_{t})$. If a reciprocal $1/m$ appears in such a hierarchy, we implicitly assume that $m$ is a positive integer. For a set $U$ and $p\in[0,1]$, a $p$-random subset of $U$ is a random subset $U^{\prime}$ of $U$ that contains each element of $U$ independently with probability $p$.

Here we briefly sketch the proofs of Theorems 1.5 and 1.6. One key idea of both proofs is that we can use the weak hypergraph regularity lemma (Theorem 2.7) to find a distribution on almost perfect matchings which has good spreadness. The weak hypergraph regularity lemma gives us a reduced $k$-uniform hypergraph $\mathcal{R}$ such that almost all subsets of $V(\mathcal{R})$ of size $d$ have large $d$-degree. Using Lemma 2.9, we can find an almost perfect matching $M_{\mathcal{R}}$ of $\mathcal{R}$ such that each edge of $M_{\mathcal{R}}$ corresponds to a vertex-disjoint pseudorandom $k$-partite subhypergraph in which we can easily construct an almost perfect matching with spreadness.

To obtain a distribution on optimal matchings, we use an approach inspired by the method of “iterative absorption” (introduced in [48, 54] and further developed in [5, 6, 7, 30, 31, 42, 56, 66]). Our iterative-absorption approach, combined with the regularity lemma, allows us to ‘bootstrap’ results on the existence of optimal matchings to construct well-spread distributions on optimal matchings. In this approach, we choose a random partition $(U_{1},\dots,U_{\ell})$ of the vertex-set of the $k$-uniform hypergraph $\mathcal{H}$, which we call a vertex vortex (Definition 3.1), which a.a.s. satisfies $|U_{i+1}|\sim|U_{i}|/2$, $|U_{\ell}|=O(n^{1/k})$, and some additional conditions on degrees of the vertices in $\mathcal{H}[U_{i}]$ and $\mathcal{H}[U_{i},U_{i+1}]$. Using the regularity-lemma approach described above, we can find a well-spread distribution on matchings in $\mathcal{H}[U_{i}]$ which cover almost all vertices. Then we cover the leftover uncovered vertices in $U_{i}$ using edges which intersect $U_{i+1}$ in $k-1$ vertices. By the degree conditions of the vertex vortex, there are many choices of such edges, so a random greedy approach yields a distribution with good enough spreadness. After iterating this procedure $\ell-1$ times, it suffices to find an optimal matching in the final subset $U_{\ell}$ (with a small subset of vertices deleted), in a deterministic way, since $|U_{\ell}|=O(n^{1/k})$ and $\mathbb{P}\left[e_{1},\dots,e_{t}\subseteq U_{\ell}\right]=(|U_{\ell}|/n)^{kt}=(O(1)/n^{k-1})^{t}$ for any $t$ disjoint edges $e_{1},\dots,e_{t}\in\mathcal{H}$.

In the setting of Theorem 1.5, it is straightforward to find an optimal matching in the final step; the hypergraph induced on the remaining vertices will still be sufficiently dense. For Theorem 1.6, we show that the result of Rödl, Ruciński, and Szemerédi [63] holds ‘robustly’. Roughly speaking, it holds in the hypergraph induced by a random set of vertices, even after deleting a small proportion of the vertices. For this we must consider two cases according to whether the original hypergraph is close to being a ‘critical hypergraph’ (see Definition 4.2) which has minimum codegree $m_{k-1}(k,n)-1$ and no perfect matching. If the original hypergraph $\mathcal{H}$ is close to being a critical hypergraph, then we may choose an ‘atypical edge’ among $\Omega(n^{k-1})$ candidates (Lemma 6.9) and delete its vertices in advance. This ensures that the subhypergraph of $\mathcal{H}$ induced by the remaining vertices in $U_{\ell}$ will meet certain ‘divisibility’ conditions and allow us to apply some technical results proved by Rödl, Ruciński, and Szemerédi to find a perfect matching covering the remaining vertices of $U_{\ell}$. Moreover, since there are $\Omega(n^{k-1})$ candidates for the atypical edge, we have the desired spreadness property for this edge. In the second case, the original hypergraph $\mathcal{H}$ is not close to being a critical hypergraph. In this case, we prove that there are still many ‘absorbers’ inside $U_{\ell}$ (Corollary 5.4), which we can use to build an ‘absorbing matching’. As long as the vertices of the absorbing matching are not among those removed from $U_{\ell}$, we can transform an almost perfect matching (which covers most of the remaining vertices of $U_{\ell}$) into a perfect matching (i.e., one which covers all remaining vertices of $U_{\ell}$).

We will use the following well-known version of the Chernoff bound.

We will use the following probabilistic lemma which follows from the Kim–Vu polynomial concentration theorem [47]. For the proof, see LABEL:{app:lemma-proofs}.

We now introduce the weak hypergraph regularity lemma, which states that any $k$-uniform hypergraph has a vertex partition into clusters $\{V_{i}\}_{0\leq i\leq t}$ so that almost all $k$-tuples of clusters induce $\varepsilon$-regular subhypergraphs. Since the notion of $\varepsilon$-regularity is ‘weak’, its proof is very similar to the graph version. Readers should not confuse the weak hypergraph regularity lemma with the Frieze–Kannan weak regularity lemma [24].

The next lemma can be proved with a straightforward adaptation of the proof of [33, Proposition 16], so we defer the proof to Appendix A.

For $1\leq d\leq k-1$, recall that $m_{d}(k,n)$ is the minimum $D$ such that every $n$-vertex $k$-uniform hypergraph with minimum $d$-degree at least $D$ has an optimal matching. Let us define

Note that $\underline{\mu_{d}}(k)\leq\overline{\mu_{d}}^{(s)}(k)$ for $0\leq s\leq k-1$, and $\underline{\mu_{k-1}}(k)=1/k$, since $m_{k-1}(k,n)=n/2-O(k)$ for large $n\in k\mathbb{N}$ and $m_{k-1}(k,n)=\lfloor n/k\rfloor$ for large $n\notin k\mathbb{N}$, as mentioned in Section 1.1. A well-known lower bound on $\underline{\mu_{d}}(k)$ is $1-(\frac{k-1}{k})^{k-d}$ (see [72, Construction 1.4]).

Now we prove the following lemma which states that if almost all $d$-tuples satisfy the degree condition for an optimal matching then there exists an almost perfect matching. We also remark that there are also similar results on almost perfect matchings [25, 26, 44].

To prove Lemma 2.9, we use the following lemma [19, Lemma 3.4].