1. Introduction
A classical problem in extremal set theory is to determine the maximum number of edges in a hypergraph with no matching of a given size. Let us write
$\nu (H)$
for the size of the largest matching in a hypergraph
$H$
. Erdős [Reference Erdős6] in 1965 made the following conjecture (known as the Erdős Matching Conjecture): For positive integers
$k, n, t$
with
$n \ge kt$
, every
$k$
-uniform hypergraph
$H$
(or
$k$
-graph for short) on
$n$
vertices with
$\nu (H) \lt t$
satisfies
$e(H)\leq \max \left \{ \binom{kt-1}{k}, \binom{n}{k}-\binom{n-t+1}{k} \right \}.$
This bound is tight for the following two extremal constructions: (a) the complete
$k$
-graph on
$kt-1$
vertices, and (b) the
$k$
-graph on
$n$
vertices in which every edge intersects a fixed set of
$t-1$
vertices. There has been recent progress on this conjecture, see [Reference Alon, Huang and Sudakov2, Reference Alon, Frankl, Huang, Rödl, Ruciński and Sudakov3, Reference Frankl, Łuczak and Mieczkowska7, Reference Frankl9, Reference Frankl10, Reference Huang, Loh and Sudakov15, Reference Łuczak and Mieczkowska20]. In particular, Frankl [Reference Frankl9] proved that if
$n\geq (2t-1)k-(t-1)$
and
$\nu (H)\lt t$
then
$e(H)\le \binom{n}{k}-\binom{n-t+1}{k}$
, with further improvement by Frankl and Kupavskii [Reference Frankl and Kupavskii11].
The work in this paper was motivated by a recent result of Frankl and Kupavskii [Reference Frankl and Kupavskii12] on a stability version of the Erdős Matching Conjecture. We postpone the detailed discussion on this motivation to Section 6, since it uses technical notation that is not required for stating and proving our main result. For now, we simply mention that we are interested in extending a result of Frankl and Kupavskii [Reference Frankl and Kupavskii12] to rainbow matchings.
Let
$\mathcal{F} = \{F_1,\ldots , F_t\}$
be a family of hypergraphs. A set of pairwise disjoint edges, one from each
$F_i$
, is called a rainbow matching for
$\mathcal{F}$
, and we say that
${\mathcal F}$
or
$\{ F_1,\ldots , F_t\}$
admits a rainbow matching. Aharoni and Howard [Reference Aharoni and Howard1] (see also Huang, Loh and Sudakov [Reference Huang, Loh and Sudakov15]) conjectured the following rainbow version of the Erdős Matching Conjecture: If
$E(F_i)\subseteq \binom{[n]}{k}$
and
$e\,(F_i)\gt \max \left \{\binom{n}{k}-\binom{n-t+1}{k},\binom{kt-1}{k}\right \}$
for
$i\in [t]$
then
$\{ F_1,\ldots , F_t\}$
admits a rainbow matching. This conjecture was confirmed for
$n\gt 3k^2t$
by Huang, Loh, and Sudakov [Reference Huang, Loh and Sudakov15]. The bound
$3k^2t$
on
$n$
was improved to
$12kt \log (e^2t)$
by Frankl and Kupavskii [Reference Frankl and Kupavskii13], and subsequently to
$3ekt$
by Kupavskii [Reference Kupavskii18]. Lu, Wang, and Yu [Reference Lu, Wang and Yu19] further improved this bound to
$n \gt 2kt$
for
$t$
sufficiently large. Recently, Keevash, Lifshitz, Long, and Minzer [Reference Keevash, Lifshitz, Long and Minzer17] proved a more general result with
$n \ge Ckt$
for some constant
$C$
using sharp threshold techniques developed in [Reference Keevash, Lifshitz, Long and Minzer16].
Frankl and Kupavskii [Reference Frankl and Kupavskii12] recently proved a result generalizing the the Hilton-Milner theorem by considering vertex covers of hypergraphs. A vertex cover of a hypergraph
$H$
is a set of vertices of
$H$
that meets all edges of
$H$
, and we use
$\tau (H)$
to denote the minimum size of a vertex cover of
$H$
. Our main result below extends that result of Frankl and Kupavskii [Reference Frankl and Kupavskii12] to rainbow matchings. (By letting
$k_i = k$
and
$F_i = F$
for all
$i \in [t+1]$
, Theorem1 gives that result of Frankl and Kupavskii.)
Theorem 1.
Let
$k\ge 1$
be an integer. There exists
$t_0=t_0(k)$
such that, for positive integers
$t, k_1,k_2,\ldots ,k_{t+1}, n$
with
$t\gt t_0$
,
$\max \{k_1, \ldots , k_{t+1}\}\le k$
, and
$n \gt 2k(t+1)$
, the following holds: If
$E(F_i) \subseteq \binom{[n]}{k_i}$
and
\begin{equation*} e\,(F_i)\gt \binom{n}{k_i}-\binom{n-t}{k_i} - \binom{n-t-k}{k_i-1} + 1 \end{equation*}
for all
$i \in [t+1]$
, then either
-
(i)
$\{F_1, \ldots , F_{t+1}\}$
admits a rainbow matching of size
$t+1$
; or
-
(ii) there exists
$W \in \binom{[n]}{t}$
such that
$W$
is a vertex cover of
$F_i$
for all
$i \in [t+1]$
.
In Section 2, we first reduce the problem for finding rainbow matchings for non-uniform hypergraphs to one for uniform hypergraphs (through an operation called ‘expansion’), and then reduce the problem to a matching problem for
${{\mathcal F}}^{t+1}(k,n)$
, a special class of uniform hypergraphs. (We will see that the extremal hypergraphs of Theorem1 naturally corresponds to a special class of
$(k+1)$
-graphs, denoted by
${{\mathcal F}}_{t+1}(n,k;\;t)$
and defined later.) This new matching problem for
${{\mathcal F}}^{t+1}(k,n)$
will be further reduced to a near perfect matching problem.
In Section 3, we prove Theorem1 for small
$t$
by an inductive argument. We prove Theorem1 in Section 4 when
${{\mathcal F}}^{t+1}(k,n)$
is close to
${{\mathcal F}}_{t+1}(k,n;\;t)$
, in the sense that most edges of
${{\mathcal F}}_{t+1}(k,n;\;t)$
are also edges of
${{\mathcal F}}^{t+1}(k,n)$
. In Section 5, we deal with the case when
${{\mathcal F}}^{t+1}(k,n)$
is not close to
${{\mathcal F}}_{t+1}(k,n;\;t)$
. In Section 6, we discuss a conjecture of Frankl and Kupavskii [Reference Frankl and Kupavskii12] that motivates our work and conclude with a rainbow version of that conjecture.
2. Reductions
The goal of this section is to reduce the problem for finding rainbow matchings for non-uniform hypergraphs to a near perfect matching problem for uniform hypergraphs. We need to use shadows of hypergraphs. Let
$k, m$
be positive integers. The
$k$
-cascade representation of
$m$
is
\begin{equation*}m = \binom{a_k}{k} + \binom{a_{k-1}}{k-1} + \cdots + \binom{a_s}{s}\end{equation*}
where
$a_k \gt a_{k-1} \gt \ldots \gt a_s \ge s \ge 1$
are integers. Given a family
${\mathcal F}$
of sets, the shadow
$\partial {{\mathcal F}}$
of
${\mathcal F}$
is defined as
\begin{equation*}\partial {{\mathcal F}} = \{E \;:\; E = F \setminus \{x\} \text{ for some } F \in {{\mathcal F}} \text{ and } x \in F \}.\end{equation*}
For
$i \ge 1$
, we define
$\partial ^{i+1} {{\mathcal F}} = \partial (\partial ^{i} {{\mathcal F}})$
. The following result is known as the Kruskal-Katona theorem (see Theorem 1 in [Reference Frankl8]).
Theorem 2 (Kruskal and Katona). Let
$n,k$
be positive integers and
${{\mathcal F}} \subseteq \binom{[n]}{k}$
. If
$|{{\mathcal F}}| = \binom{a_k}{k} + \binom{a_{k-1}}{k-1} + \cdots + \binom{a_s}{s}$
where
$a_k \gt a_{k-1} \gt \ldots \gt a_s \ge s \ge 1$
are integers. Then
\begin{equation*} |\partial {{\mathcal F}}| \ge \binom{a_k}{k-1} + \binom{a_{k-1}}{k-2} + \cdots + \binom{a_s}{s-1}. \end{equation*}
Applying Theorem2 repeatedly, we obtain the following
Corollary 3.
Let
$n,k$
be positive integers and
${{\mathcal F}} \subseteq \binom{[n]}{k}$
. If
$|{{\mathcal F}}| = \binom{a_k}{k} + \binom{a_{k-1}}{k-1} + \cdots + \binom{a_1}{1}$
where
$a_k \gt a_{k-1} \gt \ldots \gt a_1 \ge 1$
are integers. Let
$1 \le r \le k-1$
. Then
\begin{equation*} |\partial ^{k-r} {{\mathcal F}}| \ge \binom{a_k}{r} + \binom{a_{k-1}}{r-1} + \cdots + \binom{a_{k-r}}{0}. \end{equation*}
The first step of our reduction needs the following connection between the number of edges in an
$r$
-graph and its
$k$
-uniform ‘expansion’.
Lemma 4.
Let
$n \ge k \gt r \ge 1$
be integers. Let
$\mathcal{A}\subseteq \binom{[n]}{r}$
and let
$\mathcal{B}= \{e \cup f\;:\; e \in \mathcal{A}, f \in \binom{[n] \setminus e}{k-r} \}.$
If
$|\mathcal{A}|\gt \binom{n}{r}-\binom{n-t}{r} - \binom{n-t-k}{r-1} + 1$
, then
$|\mathcal{B}|\gt \binom{n}{ k}-\binom{n-t}{k} - \binom{n-t-k}{k-1} + 1$
.
Proof.
Suppose the conclusion is false. Then the number of elements in
$\binom{[n]}{k}\setminus \mathcal{B}$
, satisfies
\begin{align*} \begin{split} \left |\binom{[n]}{k}\setminus \mathcal{B}\right | &\geq \binom{n-t}{k} + \binom{n-t-k}{k-1} - 1 \\ &= \binom{n-t}{k} +\sum _{i=1}^{k-1}\binom{n-t-k-i}{k-i}, \end{split} \end{align*}
where in the second equality we repeatedly used the formula
$\binom{x}{y}=\binom{x-1}{y}+\binom{x-1}{y-1}$
to
$\binom{n-t-k}{k-1}$
.
Thus, by applying Corollary 3, we have
\begin{align*} \begin{split} \left |\partial ^{k-r} \left (\binom{[n]}{k}\setminus \mathcal{B}\right )\right | &\ge \binom{n-t}{r} +\sum _{i=1}^{r}\binom{n-t-k-i}{r-i} \\ &= \binom{n-t}{r} + \binom{n-t-k}{r-1} \end{split} \end{align*}
Since
$\partial ^{k-r} (\binom{[n]}{k}\setminus \mathcal{B}) \cap \mathcal{A} = \emptyset$
(by definition),
$|\mathcal{A}| \le \binom{n}{r}-\binom{n-t}{r} - \binom{n-t-k}{r-1}$
, a contradiction.
By Lemma 4, Theorem1 follows from the following result.
Theorem 5.
Let
$k \ge 3$
be a positive integer. There exists
$t_0=t_0(k)$
such that, for integers
$t,n$
with
$t \gt t_0$
and
$n \gt 2k(t+1)$
, if
$E(F_i) \subseteq \binom{[n]}{k}$
and
\begin{equation*} e\,(F_i)\gt \binom{n}{k}-\binom{n-t}{k} - \binom{n-t-k}{k-1} + 1, \end{equation*}
for
$i\in [t+1]$
, then
-
(i)
$\{F_1, \ldots , F_{t+1}\}$
admits a rainbow matching; or
-
(ii) there exists
$W \in \binom{[n]}{t}$
such that
$W$
is a vertex cover of
$F_i$
for all
$i\in [t+1]$
.
To prove Theorem5, we convert this rainbow matching problem on
$k$
-graphs to a matching problem for a special class of
$(k+1)$
-graphs. Let
$F_1,\ldots , F_t$
be a family of
$k$
-graphs and
$X\;:\!=\;\{x_1,\ldots ,x_t\}$
be a set of
$t$
vertices disjoint from
$[n]$
. We use
$\mathcal{F}^t(k,n)$
to denote the
$(k+1)$
-graph with vertex set
$X\cup [n]$
and edge set
\begin{equation*} E(\mathcal{F}^t(k,n))=\bigcup _{i=1}^t \{\{x_i\}\cup e\ :\ e\in F_i\}. \end{equation*}
Observation 6.
$\{F_1,\ldots ,F_t\}$
admits a rainbow matching if, and only if,
$\mathcal{F}^t(k,n)$
has a matching of size
$t$
.
Hence, to prove Theorem5, we need to see when
$\mathcal{F}^{t+1}(k,n)$
has a matching of size
$t+1$
. We further reduce this problem to a near perfect matching problem.
Write
$n-kt=km+r$
, where
$0\leq r\leq k-1$
. Let
$F_1, \ldots , F_t\subseteq \binom{[n]}{k}$
, and let
$F_i=\binom{[n]}{k}$
for
$i=t+1, \ldots , t+m$
. Let
$Q=\{x_1,\ldots ,x_{m+t}\}$
disjoint from
$[n]$
and let
$\mathcal{H}^{t}(k,n)$
be the
$(k+1)$
-graph with vertex set
$Q\cup [n]$
and edge set
\begin{align*} E(\mathcal{H}^{t}(k,n))=\left ( \bigcup _{i=1}^{t}\{\{x_i\}\cup e\ :\ e\in F_i\} \right ) \bigcup \left ( \bigcup _{i=t+1}^{t+m}\left\{\{x_i\}\cup e\ :\ e\in \binom{[n]}{k} \right\} \right ). \end{align*}
When
$F_1=\cdots =F_{t}=H_k(s,n)$
, we denote
$\mathcal{H}^{t}(k,n)$
by
$\mathcal{H}_{t}(k,n;\;s)$
. Note that
$\nu (\mathcal{H}_{t}(k,n;\;t))=t+m=(n-r)/k$
, i.e.,
${\mathcal H}_t(k,n;\;t)$
has a matching covering all but at most
$r\lt k$
vertices (and such a matching is said to be near perfect).
The following lemma provides an equivalent condition on matchings in
$\mathcal{F}^{t}(k,n)$
and matchings in
$\mathcal{H}_{t}(k,n;\;t)$
. See Lemma 2.1 in [Reference Lu, Wang and Yu19].
Lemma 7.
Let
$n,k,t$
be positive integers and let
$F_1, \ldots , F_{t}\subseteq \binom{[n]}{k}$
. Then
$\mathcal{F}^{t}(k,n)$
has a matching of size
$t$
if, and only if,
$\mathcal{H}_t(k,n;\;t)$
has a matching of size
$m+t$
, where
$m+t=\lfloor n/k\rfloor$
.
For the proof of Theorem5, we need to tell how close a hypergraph is to another hypergraph. Given two
$k$
-hypergraphs
$H_1, H_2$
with
$V(H_1)=V(H_2)$
, let
$c(H_1,H_2)$
be the minimum of
$|E(H_1)\backslash E(H')|$
taken over all isomorphic copies
$H'$
of
$H_2$
with
$V(H') = V(H_2)$
. For a real number
$\varepsilon \gt 0$
, we say that
$H_2$
is
$\varepsilon$
-close to
$H_1$
if
$V(H_1) = V(H_2)$
and
$c(H_1,H_2)\leq \varepsilon |V(H_1)|^k$
. If
$F_1=\cdots =F_t=H_k(s,n)$
, where
$1\le s\lt n$
and
$H_k(s,n)$
denotes the
$k$
-graph with vertex set
$[n]$
and edge set
$\binom{[n]}{k}\setminus \binom{[n]\setminus [s]}{k}$
, then we denote such
$\mathcal{F}^t(k,n)$
by
$\mathcal{F}_t(k,n;\;s)$
. The following is straightforward to verify.
Observation 8.
(i) If
$\mathcal{F}^t(k,n)$
is
$\varepsilon$
-close to
$\mathcal{F}_t(k,n;\;s)$
then
$\mathcal{H}^t(k,n)$
is
$\varepsilon$
-close to
$\mathcal{H}_t(k,n;\;s)$
; (ii) For
$n \le 2k^3\min \{t,s\}$
, if
$\mathcal{H}^t(k,n)$
is
$\varepsilon$
-close to
$\mathcal{H}_t(k,n;\;s)$
then
$\mathcal{F}^t(k,n)$
is
$6 \varepsilon$
-close to
$\mathcal{F}_t(k,n;\;s)$
.
Proof. We first prove (i). By definition, we have
\begin{align} |E(\mathcal{F}_t(k,n;\;s))\setminus E(\mathcal{F}^t(k,n))|\leq \varepsilon |V(\mathcal{F}^t(k,n))|^{k+1}. \end{align}
Thus, we can infer that
\begin{align*} |E(\mathcal{H}_t(k,n;\;s))\setminus E( \mathcal{H}^t(k,n))|&=|E(\mathcal{F}_t(k,n;\;s))\setminus E(\mathcal{F}^t(k,n))|\\ & \leq \varepsilon |V(\mathcal{F}^t(k,n))|^{k+1}\quad \mbox{(by (1))}\\ &\leq \varepsilon |V(\mathcal{H}^t(k,n))|^{k+1}. \end{align*}
It follows that
$\mathcal{H}^t(k,n)$
is
$\varepsilon$
-close to
$\mathcal{H}_t(k,n;\;s)$
.
Next, we prove (ii). By definition, we have
\begin{align} |E(\mathcal{H}_t(k,n;\;s))\setminus E( \mathcal{H}^t(k,n))|\leq \varepsilon |V(\mathcal{H}^t(k,n))|^{k+1}. \end{align}
One can that
\begin{align} |V(\mathcal{H}^t(k,n))|\leq n+n/k. \end{align}
Thus we can get
\begin{align*} |E(\mathcal{F}_t(k,n;\;s))\setminus E(\mathcal{F}^t(k,n))|&=|E(\mathcal{H}_t(k,n;\;s))\setminus E( \mathcal{H}^t(k,n))|\\ & \leq \varepsilon (n+n/k)^{k+1}\quad \mbox{(by (2) and (3) )}\\ &\leq 6\varepsilon n^{k+1}. \end{align*}
So it follows that
$\mathcal{F}^t(k,n)$
is
$6\varepsilon$
-close to
$\mathcal{F}_t(k,n;\;s)$
. This completes the proof of Observation 8.
Our proof of Theorem5 will be divided into two parts, according to whether or not
$\mathcal{F}^{t+1}(k,n)$
is close to
$\mathcal{F}_{t+1}(n,k;\;t)$
. If
$\mathcal{F}^{t+1}(k,n)$
is close to
$\mathcal{F}_{t+1}(n,k;\;t)$
, we will apply greedy argument to construct a matching of size
$t$
. If
$\mathcal{F}^{t+1}(k,n)$
is not close to
$\mathcal{F}_{t+1}(n,k;\;t)$
, then by Observation 8,
$\mathcal{H}^{t+1}(k,n)$
is not close to
$\mathcal{H}_{t+1}(n,k;\;t)$
(see proof of Theorem5), and we will show that
$\mathcal{H}^{t+1}(k,n;\;t)$
has a small matching
$M_1$
with nice absorbing properties and
$\mathcal{H}^{t+1}(k,n;\;t)-V(M_1)$
has a spanning subgraph in which we can find a large matching
$M_2$
that can be extended to a near perfect matching by using
$M_1$
.
3. Stability results on small matchings
We begin with the result of Huang, Loh, and Sudakov [Reference Huang, Loh and Sudakov15] mentioned previously, and state a corollary which generalizes it to non-uniform hypergraphs.
Theorem 9 (Huang, Loh, and Sudakov). Let
$n,k,t$
be three positive integers such that
$n \gt 3k^2t$
. Let
${{\mathcal F}}=\{F_1,\ldots , F_t\}$
be a family of
$k$
-graphs. If
\begin{equation*} e\,(F_i)\gt \binom{n}{k}-\binom{n-t+1}{k} \end{equation*}
for all
$1\leq i\leq t$
, then
${\mathcal F}$
admits a rainbow matching.
Corollary 10.
Let
$t,n,k_1,\ldots ,k_t,k$
be positive integers such that
$k = \max \{k_1,\ldots ,k_t\}$
and
$n \gt 3k^2t$
. Let
${{\mathcal F}}=\{F_1,\ldots , F_t\}$
be a family where
$E(F_i)\subseteq \binom{[n]}{k_i}$
for
$i \in [t]$
. If
\begin{equation*} e\,(F_i)\gt \binom{n}{k_i}-\binom{n-t+1}{k_i} \end{equation*}
for all
$1\leq i\leq t$
, then
${\mathcal F}$
admits a rainbow matching.
Proof.
For
$i \in [t]$
, we define the expansion of
$E(F_i)$
as
\begin{equation*}E(G_i) = \left\{e \cup f\;:\; e \in F_i, f \in \binom{[n] \setminus e}{k-k_i} \right\}.\end{equation*}
Note that each
$G_i$
is a
$k$
-graph.
We claim that
$|\binom{[n]}{k}\setminus E(G_i)|\lt \binom{n-t+1}{k}$
for all
$i\in [t]$
. For, otherwise, there exists some
$i\in [t]$
such that
$|\binom{[n]}{k}\setminus E(G_i)|\ge \binom{n-t+1}{k}$
. Then by Theorem2,
$|\partial ^{k-k_i} \big(\binom{[n]}{k}\setminus E(G_i)\big)| \ge \binom{n-t+1}{k_i}$
. Since
$\partial ^{k-k_i} \big(\binom{[n]}{k}\setminus E(G_i)\big) \cap E(F_i) = \emptyset$
(by definition),
$e\,(F_i) \le \binom{n}{k_i}-\binom{n-t+1}{k_i}$
, a contradiction.
Therefore,
$e(G_i)\gt \binom{n}{k}-\binom{n-t+1}{k}$
. Hence, by Theorem9,
$\{G_1,\ldots , G_t\}$
admits a rainbow matching, which implies that
${\mathcal F}$
admits a rainbow matching.
We also need a classical theorem of Mörs, which is Theorem 6 in [Reference Mörs21].
Theorem 11 (Mörs, [Reference Mörs21]). Let
$n,k,l$
be three positive integers such that
$n \ge k+l$
. Let
$\mathcal{A} \in \binom{[n]}{k}$
and
$\mathcal{B} \in \binom{[n]}{l}$
. Suppose
$\bigcap _{C \in \mathcal{A} \cup \mathcal{B}} C = \emptyset$
and
$A \cap B \ne \emptyset$
for all
$A \in \mathcal{A}$
and
$B \in \mathcal{B}$
. Then either
\begin{equation*}|\mathcal{A}| \le \binom{n-1}{k-1} - \binom{n-1-l}{k-1} + 1\end{equation*}
or
\begin{equation*}|\mathcal{B}| \le \binom{n-1}{l-1} - \binom{n-1-k}{l-1} + 1.\end{equation*}
To prove Theorem1, we need to establish the following result, which may be viewed as a stability version (as well as a generalization) of the above mentioned result of Huang, Loh, and Sudakov [Reference Huang, Loh and Sudakov15] as well as an earlier result of Bollobás, Daykin, and Erdős [Reference Bollobás, Daykin and Erdős5]. Note the following theorem holds for every positive integer
$t$
.
Theorem 12.
Let
$n, t, k_1,k_2,\ldots ,k_{t+1}$
be positive integers such that
$k_1 \ge k_2 \ge \ldots \ge k_{t+1} \ge 2$
and
$n \ge 2 k_1^2 k_2 t$
. Let
$E(F_i)\subseteq \binom{[n]}{k_i}$
,
$i \in [t+1]$
, such that, for
$i\in [2]$
,
\begin{equation*}e\,(F_i) \gt \binom{n}{k_i} - \binom{n-t}{k_i} - \binom{n-t-k_{3-i}}{k_i-1} + 1,\end{equation*}
and for
$i\in [t+1]\setminus [2]$
,
\begin{equation*}e\,(F_i)\gt \binom{n}{k_i} - \binom{n-t}{k_i} - \binom{n-t-k_{3}}{k_i-1} + 1.\end{equation*}
Then one of the following holds:
-
(i)
$\{F_1, \ldots , F_{t+1}\}$
admits a rainbow matching of size
$t+1$
; or
-
(ii) there exists
$W \in \binom{[n]}{t}$
such that
$W$
is a vertex cover of
$F_i$
for all
$i \in [t+1]$
.
Proof.
We apply induction on
$t$
. For the base case, suppose
$t=1$
. Then for
$i\in [2]$
,
\begin{equation*}e\,(F_i) \gt \binom{n}{k_i}-\binom{n-1}{k_i}-\binom{n-1-k_{3-i}}{k_i-1}+1=\binom{n-1}{k_i-1}-\binom{n-1-k_{3-i}}{k_i-1}+1;\end{equation*}
so (i) of Theorem12 follows from Theorem11.
Now suppose
$t\ge 2$
and the conclusion holds with
$t$
hypergraphs. Moreover, we may assume (i) does not hold, i.e.,
$\{F_1, \ldots , F_{t+1}\}$
does not admit a rainbow matching.
Since
$k_1\ge k_i$
for
$i\in [t+1]$
, we have from the assumption of Theorem12 that
\begin{align} e\,(F_i) \ge \binom{n}{k_i}-\binom{n-t}{k_i}-\binom{n-t-k_1}{k_i-1}+1 \quad \text{for} \, i\in [t+1]. \end{align}
Hence, since
$\binom{n-t}{k_i-1} \ge \binom{n-t-k_j}{k_i-1}$
and
$\binom{n-(t-1)}{k_i}=\binom{n-t}{k_i}+\binom{n-t}{k_i-1}$
,
\begin{equation*} e\,(F_i)\gt \binom{n}{k_i} - \binom{n-(t-1)}{k_i} \quad \text{for} \, i \in [t]. \end{equation*}
Thus by Corollary 10,
$\{F_1,\ldots ,F_t\}$
admits a rainbow matching, say
$M_1$
. Therefore, since
$\{F_1, \ldots , F_{t+1}\}$
does not admit a rainbow matching, every edge in
$F_{t+1}$
must intersect
$V(M_1)$
. Thus the maximum degree
$\Delta (F_{t+1}) \ge e(F_{t+1}) / |V(M_1)|$
. Let
$v \in [n]$
such that
$d_{F_{t+1}}(v) = \Delta (F_{t+1})$
. Then, since
\begin{align*} \binom{n}{k_{i}}&=\binom{n-1}{k_{i}-1}+\binom{n-1}{k_{i}} =\sum _{i=1}^{t-1}\binom{n-i}{k_{i}-1} +\binom{n-t}{k_{i}} \ge (t-1)\binom{n-t}{k_{i}-1} +\binom{n-t+1}{k_{i}}, \end{align*}
then by (4), we have
\begin{align*} e\,(F_i) \ge (t-1)\binom{n-t}{k_{i}-1}\quad \text{for} \,i\in [t+1]. \end{align*}
Thus if follow that
\begin{align*} d_{F_{t+1}}(v)& \ge \frac {e(F_{t+1})}{|V(M_1)|} \gt \frac {(t-1)\binom{n-t}{k_{t+1}-1}}{\sum _{i=1}^{t} k_i}. \end{align*}
Hence, since
$n \ge 2 k_1^2 k_2 t$
and
$k_1\geq k_2\geq \max \{k_i\ |\ 3\leq i\leq t+1\}$
,
\begin{align*} d_{F_{t+1}}(v) &\gt \frac {(t-1)\binom{n-t}{k_{t+1}-1}}{\sum _{i=1}^{t} k_i}\\ &\geq \frac {n(t-1)(1-1/2k^2_{1}k_2)^{k_{t+1}-1}\binom{n}{k_{t+1}-2}}{(k_{t+1}-1)\sum _{i=1}^{t} k_i}\\ &\gt (1-\frac {k_{t+1}-1}{2k_1^2k_2})\frac {n(t-1)\binom{n}{k_{t+1}-2}}{(k_{t+1}-1)\sum _{i=1}^{t} k_i}\\ &\gt \frac {n(t-1)\binom{n}{k_{t+1}-2}}{k_t\sum _{i=1}^{t} k_i}\\ & \gt \left (\sum _{i=1}^{t} k_i \right ) \binom{n-2}{k_{t+1} - 2}. \end{align*}
If
$\{F_1 - v, \ldots , F_t - v\}$
admits a rainbow matching, say
$M_2$
, then
$d_{F_{t+1}}(v)\gt |V(M_2)| \binom{n-2}{k_{t+1} - 2}$
as
$|V(M_2)| = \sum _{i=1}^{t} k_i$
. So there exists an edge
$e \in F_{t+1}$
such that
$v \in e$
and
$e \cap V(M_2) =\emptyset$
. Now
$M_2 \cup \{e\}$
is a desired rainbow matching for
$\{F_1, \ldots , F_t, F_{t+1}\}$
, a contradiction.
So
$\{F_1 - v, \ldots , F_t - v\}$
does not admit any rainbow matching. Note that, for
$i\in [2]$
,
\begin{equation*}e(F_i - v)\! \ge\! e\,(F_i)\! -\! \binom{n-1}{k_i - 1} \!\gt\! \binom{n-1}{k_i}\! - \!\binom{(n-1)-(t-1)}{k_i}\! -\! \binom{(n-1)-(t-1)-k_{3-i}}{k_i-1}\! +\! 1,\end{equation*}
and, for
$i\in [t]\setminus [2]$
,
\begin{equation*}e(F_i - v) \ge e\,(F_i) - \binom{n-1}{k_i - 1} \gt \binom{n-1}{k_i} - \binom{(n-1)-(t-1)}{k_i} - \binom{(n-1)-(t-1)-k_{3}}{k_i-1} + 1.\end{equation*}
Thus, by inductive hypothesis (applied to
$\{F_1 - v, \ldots , F_t - v\}$
), there exists
$W' \in \binom{[n] \setminus \{v\}}{t-1}$
such that
$W'$
is a vertex cover of
$F_i-v$
for all
$i \in [t]$
.
Write
$W = W' \cup \{v\}$
. We may assume that there exists
$f \in F_{t+1}$
such that
$f \cap W = \emptyset$
; for otherwise (ii) holds. Note that the number of edges in
$F_i$
(for each
$i\in [t]$
) intersecting both
$f$
and
$W$
is
\begin{equation*}e_i(f, W) \le \binom{n}{k_i} - \binom{n-k_{t+1}}{k_i} - \binom{n-t}{k_i} + \binom{n-t-k_{t+1}}{k_i}.\end{equation*}
Hence, for each
$i \in [t]$
, since every edge of
$F_i$
intersects
$W$
, the number of edges in
$F_i$
disjoint from
$f$
is
\begin{align*} \begin{split} e(F_i - f) &= e\,(F_i) - e_i(f,W) \\ &\quad \gt \left(\binom{n}{k_i} - \binom{n-t}{k_i} - \binom{n-t-k_{3}}{k_i-1} + 1 \right ) -e(f,W) \\ &\quad\ge \binom{n-k_{t+1}}{k_i} - \binom{n-t-k_{3}}{k_i-1} - \binom{n-t-k_{t+1}}{k_i} + 1 \\ &\quad\gt \binom{n-k_{t+1}}{k_i} - \binom{(n-k_{t+1})-(t-1)}{k_i} \end{split} \end{align*}
Hence by Corollary 10,
$\{F_1-V(f), F_2-V(f), \ldots , F_t-V(f)\}$
, i.e. deleting all vertices in
$f$
, admits a rainbow matching
$M_3$
of size
$t$
. Therefore,
$M_3 \cup \{f\}$
is a rainbow matching of size
$t+1$
which satisfies (i).
As a consequence of Theorem12, we have the following conclusion.
Corollary 13.
Let
$t, k_1,k_2,\ldots ,k_{t+1}, n$
be positive integers, let
$k = \max \{k_1, \ldots , k_{t+1}\}$
, and let
$E(F_i) \subseteq \binom{[n]}{k_i}$
for
$i \in [t+1]$
. Suppose
$n \ge 2 k^3t$
and
$e\,(F_i) \gt \binom{n}{k_i} - \binom{n-t}{k_i} - \binom{n-t-k}{k_i-1} + 1$
for all
$i \in [t+1]$
. Then
-
(i)
$ \{F_1,F_2,\cdots ,F_{t+1}\}$
admits a rainbow matching of size
$t+1$
; or
-
(ii) there exists
$W\in \binom{[n]}{t}$
such that
$W$
is a vertex cover of
$F_i$
for all
$i \in [t+1]$
.
By taking
$k_1=k_2=\cdots =k_{t+1}=k$
in Corollary 13, we have the following
Corollary 14.
Let
$t,k,n$
be positive integers and let
$E(F_i) \subseteq \binom{[n]}{k}$
for
$i \in [t+1]$
. Suppose
$n \ge 2 k^3t$
and
$e\,(F_i) \gt \binom{n}{k} - \binom{n-t}{k} - \binom{n-t-k}{k-1} + 1$
for all
$i \in [t+1]$
. Then either
-
(i)
$ \{F_1,F_2,\cdots ,F_{t+1}\}$
admits a rainbow matching of size
$t+1$
; or
-
(ii) there exists
$W\in \binom{[n]}{t}$
such that
$W$
is a vertex cover of
$F_i$
for all
$i \in [t+1]$
.
4. Extremal case
Recall
$\mathcal{F}^{t+1}(k,n)$
is the
$(k+1)$
-graph with vertex set
$X\cup [n]$
and edge set
$E(\mathcal{F}^{t+1}(k,n))=\bigcup _{i=1}^{t+1} \{\{x_i\}\cup e\ :\ e\in F_i\}$
. We also recall that
$\mathcal{F}_{t+1}(k,n;\;s)$
is the
$(k+1)$
-graph with vertex set
$X\cup [n]$
and edge set
$E(\mathcal{F}_{t+1}(k,n;\;s))=\bigcup _{i=1}^{t+1} \{\{x_i\}\cup e\ :\ e\in H_k(s,n)\}$
where
$H_k(s,n)$
denotes the
$k$
-graph with vertex set
$[n]$
and edge set
$\binom{[n]}{k}\setminus \binom{[n]\setminus [s]}{k}$
.
In this section, we prove Theorem1 for the case when
$\mathcal{F}^{t+1}({k},n)$
is
$\varepsilon$
-close to the extremal configuration
$\mathcal{F}_{t+1}({k},n;\;t)$
for some
$ \varepsilon \ll 1/k$
. We write
$a\ll b$
to mean that there exists an increasing function
$f$
such that our result holds whenever
$a\leq f(b) \lt b$
.
Let
$H$
be a
$(k+1)$
-graph and
$v \in V(H)$
. We define the neighborhood
$N_H(v)$
of
$v$
in
$H$
to be the set
$ \{ S \in \binom{V(H)}{k} \ :\ S \cup \{v\} \in E(H) \}$
. Let
$H$
be a
$(k+1)$
-graph with the same vertex set as
$\mathcal{F}_{t+1}(k,n;\;t)$
. Given real number
$\alpha$
with
$0\lt \alpha \lt 1$
, a vertex
$v$
in
$H$
is called
$\alpha$
-good with respect to
$\mathcal{F}_{t+1}(k,n;\;t)$
if
\begin{equation*}\left |N_{\mathcal{F}_{t+1}(k,n;\;t)}(v)\setminus N_H(v)\right |\le \alpha n^{k}.\end{equation*}
Clearly, if
$H$
is
$\varepsilon$
-close to
$\mathcal{F}_{t+1}(k,n;\;t)$
, then at most
$(k+1)(1+1/k)^{k+1}\varepsilon n/\alpha$
vertices of
$H$
are not
$\alpha$
-good with respect to
$\mathcal{F}_{t+1}(k,n;\;t)$
.
First we deal with the case when all vertices of
$\mathcal{F}^{t+1}({k},n)$
are
$\alpha$
-good with respect to
$\mathcal{F}_{t+1}(k,n;\;t)$
. Let
$Q,V$
be two disjoint sets. A
$(k+1)$
-graph
$H$
with vertex set
$Q \cup V$
is called
$(1,k)$
-partite with partition classes
$Q,V$
if, for each edge
$e\in E(H)$
,
$|e\cap Q|=1$
and
$|e\cap V|=k$
. Clearly,
$\mathcal{F}^{t+1}({k},n)$
and
$\mathcal{F}_{t+1}({k},n;\;t)$
are
$(1,k)$
-partite graphs with partition classes
$X, [n]$
.
Lemma 15.
Let
$\zeta , \alpha$
be real numbers and
$n,k,t$
be positive integers such that
$0\lt \alpha \ll \zeta \lt 1$
,
$\alpha \ll 1/k$
,
$n \ge 8 k^4$
,
$t \ge n/(2k^3)$
and
$t+1\lt (1-\zeta )n/k$
. Let
$H$
be a
$(1,k)$
-partite
$(k+1)$
-graph with the same partition classes as
$\mathcal{F}_{t+1}(k,n;\;t)$
. If every vertex of
$H$
is
$\alpha$
-good with respect to
$\mathcal{F}_{t+1}(k,n;\;t)$
, then
$\nu (H)\geq t$
with equality only if
$H$
is a subgraph of
$\mathcal{F}_{t+1}(k,n;\;t)$
.
Proof.
Let
$X \;:\!=\; \{x_1, x_2, \ldots , x_{t+1}\}$
,
$W \;:\!=\; [t]$
, and
$U \;:\!=\; [n] \setminus [t]$
, such that
$X,[n]$
are the partition classes of
$H$
and
$\mathcal{F}_{t+1}(k,n;\;t)$
.
If
$H$
is a subgraph of
$\mathcal{F}_{t+1}(k,n;\;t)$
, then we define
$e_0 = \emptyset$
. Otherwise, there exists an edge
$e_0$
in
$H$
such that
$|e_0 \cap X| = 1$
and
$e_0\cap W=e_0 \cap [t] = \emptyset$
. Without loss of generality, we may assume
$e_0 \cap X = \{ x_{t+1} \}$
.
Write
$H' = H - V(e_0)$
, i.e. deleting all vertices in
$e_0$
. Let
$M$
be a maximum matching in
$H'$
such that
$|e\cap X| = |e\cap W|=1$
for all
$e\in M$
. Thus
$|M|\le |W|=t$
. Let
$X'=X\setminus (V(M) \cup V(e_0))$
,
$W'=W\setminus (V(M) \cup V(e_0))$
, and
$U'=U\setminus (V(M) \cup V(e_0))$
. Thus
$|W'|=|W|-|M|$
and
$|U'|\ge |U|-|M|(k-1)-k$
.
We claim that
$|M|\geq n/(4k^3)$
. For, suppose
$|M| \lt n/(4k^3)$
. Consider any vertex
$x\in X'$
. Since
$x$
is
$\alpha$
-good with respect to
$\mathcal{F}_{t+1}(k,n;\;t)$
, we have
\begin{equation*} \left | \left (W\times \binom{U}{k-1}\right )\setminus N_H(x)\right |\leq \alpha n^k. \end{equation*}
Since
$t + 1 \lt (1-\zeta )n/k$
,
$|U'|\ge |U|-|M|(k-1)-k\ge (n-t)-t(k-1)-k\gt \zeta n$
. Hence, since
$t \ge n/2k^3$
,
\begin{equation*} \left |W'\times \binom{U'}{k-1}\right |\ge (|W|-|M|)\binom{|U|-|M|(k-1)-k}{k-1}\gt \frac {n}{4k^3}\binom{\zeta n}{k-1} \gt \frac {n}{4k^3} \frac {(\zeta n/2)^{k-1}}{(k-1)!}, \end{equation*}
Therefore,
$\left |W'\times \binom{U'}{k-1}\right | \gt \alpha n^k$
, as
$\alpha \lt \zeta ^{k-1}(k^3 2^{k+1} (k-1)!)^{-1}$
. Hence, there exists
$f\in N_{H}(x)\cap \left (W'\times \binom{U'}{k-1}\right )$
. Let
$f'=\{x\}\cup f$
, Then
$f'\in E(H)$
,
$|f'\cap X|=|f'\cap W|=1$
, and
$f'\cap V(M)=\emptyset$
. Now
$M'=M\cup \{f'\}$
is a matching of size
$|M|+1$
in
$H$
, and
$|e\cap X|= |e\cap W|=1$
for all
$e\in M'$
. Thus,
$M'$
contradicts the choice of
$M$
, completing the proof of the claim.
Let
$\{u_1,\ldots ,u_{k+1}\}\subseteq V(H)\setminus V(M)$
, where
$u_1\in X'$
,
$u_{k+1}\in W'$
and
$u_i\in U'$
for
$i\in [k]\setminus \{1\}$
. Since
$|M| \ge n/(4k^3) \ge 2k$
, let
$\{e_1,\ldots ,e_{k}\}$
be an arbitrary
$k$
-subset of
$M$
, and let
$e_i \;:\!=\; \{v_{i,1},v_{i,2},\ldots ,v_{i,k+1}\}$
with
$v_{i,1} \in X$
,
$v_{i,k+1}\in W$
, and
$v_{i,j}\in U$
for
$i \in [k]$
and
$j \in [k] \setminus \{1\}$
. For
$j \in [k+1]$
, let
$f_j \;:\!=\; \{u_{j}, v_{1,j+1}, v_{2,j+2},\ldots ,$
$v_{k,j+k}\}$
with addition in the subscripts modulo
$k+1$
(except we write
$k+1$
instead of
$0$
). Note that
$f_1, \ldots , f_{k+1}$
are pairwise disjoint.
If
$f_j \in E(H)$
for all
$j \in [k+1]$
then
$M'\;:\!=\; (M \cup \{f_1,\ldots ,f_{k+1}\})\setminus \{e_1,\ldots ,e_{k}\}$
is a matching in
$H$
such that
$|M'| = |M| + 1 \gt |M|$
and
$|f\cap X|=|f\cap W|=1$
for all
$f\in M'$
, contradicting the choice of
$M$
. Hence,
$f_j\not \in E(H)$
for some
$j \in [k+1]$
.
Note that there are
$\binom {|M|}{k}k!$
choices of ordered
$k$
-tuples
$(e_1,\ldots , e_{k}) \in M^k$
and that any two different such choices correspond to different
$f_j$
. Hence,
\begin{eqnarray*} & & |\{f \in E(\mathcal{F}_{t+1}(k,n;\;t)) \setminus E(H)\;:\; |f\cap \{u_i\;:\; i\in [k+1]\}|=1\}|\\ &\geq & |M|(|M| - 1) \cdots (|M| - k +1) \\ &\gt & \left (n/(4k^3) - k \right )^{k} \\ &\gt & \left (n/(8k^3)\right )^{k} \quad (\text{since}\;n\ge 8k^4)\\ &\gt & (k+1) \alpha n^{k} \quad (\text{since}\;\alpha \lt ((k+1)8^{k}k^{3k}))^{-1}. \end{eqnarray*}
This implies that there exists
$i \in [k+1]$
such that
$|N_{\mathcal{F}_{t+1}(k,n)}(u_i) \setminus N_{H}(u_i)| \gt \alpha n^{k}$
, contradicting the fact that all
$u_i$
’s are
$\alpha$
-good with respect to
$\mathcal{F}_{t+1}(k,n;\;t)$
.
Therefore,
$H$
has a matching
$M$
of size
$t$
. Moreover,
$M \cup \{e_0\}$
is a matching of size
$t+1$
unless
$e_0=\emptyset$
in which case
$H$
is a subgraph of
$\mathcal{F}_{t+1}(k,n;\;t)$
.
We can now prove Theorem1 when
$\mathcal{F}^{t+1}(k,n)$
is
$\varepsilon$
-close to
$\mathcal{F}_{t+1}(k,n;\;t)$
.
Lemma 16.
Let
$k\ge 3$
,
$t\ge 1$
and
$n$
be integers, and let
$\varepsilon , \zeta$
be real numbers such that
$0 \lt 1/n\ll \varepsilon \ll \zeta \ll 1/k \lt 1$
,
$t + 1\lt (1-\zeta )(1-4k^2\sqrt {\varepsilon }) n/k$
. Let
${{\mathcal F}}=\{F_1, \ldots , F_{t+1}\}$
be a family of
$k$
-graphs such that
$e\,(F_i)\gt \binom{n}{k}-\binom{n-t}{k} - \binom{n-t-k}{k-1} + 1$
for
$i\in [t+1]$
. Suppose
$\mathcal{F}^{t+1}(k,n)$
is
$\varepsilon$
-close to
$\mathcal{F}_{t+1}(k,n;\;t)$
. Then
$\mathcal{F}^{t+1}(k,n)$
has a matching of size
$t+1$
or
$\mathcal{F}^{t+1}(k,n)$
is a subgraph of
$\mathcal{F}_{t+1}(k,n;\;t)$
.
Proof.
We may assume
$n\le 2k^3t$
as otherwise the assertion follows from Corollary 13. Let
$X, [n]$
be the partition classes of
${{\mathcal F}}_{t+1}(k,n;\;t)$
, and let
$X \;:\!=\; \{x_1, x_2, \ldots , x_{t+1}\}$
. Note that each edge of
${{\mathcal F}}_{t+1}(k,n;\;t)$
intersects
$[t]$
.
Let
$B$
denote the set of vertices in
$\mathcal{F}^{t+1}(k,n)$
that are not
$\sqrt {\varepsilon }$
-good with respect to
$\mathcal{F}_{t+1}(k,n;\;t)$
. Since
$\mathcal{F}^{t+1}(k,n)$
is
$\varepsilon$
-close to
$\mathcal{F}_{t+1}(k,n;\;t)$
,
\begin{equation*} |B\cap X|\leq (1+1/k)^{k+1}\sqrt {\varepsilon }n\leq 4\sqrt {\varepsilon }n, \end{equation*}
and
\begin{equation*} |B\cap [n]|\leq k(1+1/k)^{k+1}\sqrt {\varepsilon }n\leq 4k\sqrt {\varepsilon }n. \end{equation*}
Let
$b\;:\!=\;\max \{|B\cap X|, |B\cap [t]|\}$
; so
$b \le 4k\sqrt {\varepsilon }n$
. We choose
$X_1 \subseteq X$
and
$W_1 \subseteq [t]$
such that
$B\cap X \subseteq X_1$
,
$B\cap [t]\subseteq W_1$
, and
$|X_1|=|W_1|=b$
.
For each
$x_{i} \in X \setminus X_1$
,
$x_i$
is
$\sqrt {\varepsilon }$
-good with respect to
$\mathcal{F}_{t+1}(k,n;\;t)$
and we let
$\mathcal{F}_i=\mathcal{F}^{t+1}(k,n)[X_1\cup W_1\cup U \cup \{x_{i}\}]$
. For every
$x\in X_1 \cup \{x_{i}\}$
, there exists
$j\in [t+1]$
such that
$x=x_j$
, and we have
\begin{align*} \begin{split} |N_{\mathcal{F}_i}(x)| &\geq {|N_{\mathcal{F}^{t+1}(k,n)}(x_j)|}-\left (\binom {n}{k}-\binom{n-|[t]\setminus W_1|}{k}\right ) \\ &=e(F_j)-\left (\binom{n}{k}-\binom{n-(t-b)}{k}\right ) \\ &\quad \gt \binom{n-(t-b)}{k}-\binom{n-t}{k}-\binom{n-t-k}{k-1} + 1 \\ &=\binom{n-(t-b)}{k}-\binom{(n-(t-b))-b}{k} -\binom{(n-(t-b)) - b -k}{k-1} + 1. \end{split} \end{align*}
Since
$n-(t-b) \gt n/2 \ge 2k^3(k+1)\sqrt {\varepsilon }n \gt 2k^3 b$
, it follows from Corollary 13 that the family
$\{N_{\mathcal{F}_i}(x)\;:\; x\in X_1 \cup \{x_{i}\}\}$
admits a rainbow matching of size
$b+1$
, or there exists
$W^i \in \binom{[n] \setminus ([t]\setminus W_1)}{b}$
such that
$W^i$
is a vertex cover of
$N_{\mathcal{F}_i}(x)$
for all
$x\in X_1\cup \{x_i\}$
.
Suppose there exists
$x_i \in X\setminus X_1$
such that
$\{N_{\mathcal{F}_i}(x)\;:\; x\in X_1 \cup \{x_{i}\}\}$
admits a rainbow matching
$M_i$
of size
$b+1$
. Write
$H \;:\!=\; \mathcal{F}^{t+1}(k,n)[X \cup [n] \setminus (V(M_i) \cup B)]$
. Let
$n' = |V(H) \cap [n]|$
. We relabel vertices so that
$[t]\setminus W_1$
corresponds to
$[t-b]$
. Since
$b\le 4k\sqrt {\varepsilon }n$
and
$\varepsilon \ll 1/k$
, for every
$y\in V(H)$
, we have
\begin{align*} |N_{\mathcal{F}_{t-b}(k,n';\;t-b)}(y)\backslash N_{H}(y)|&\leq |N_{\mathcal{F}_{t+1}(k,n;\;t)}(y)\backslash N_{\mathcal{F}^{t+1}(n,b)}(y)|\\ &\leq \sqrt {\varepsilon }(n+t+1)^k\\ &\lt \varepsilon ^{1/3}(n/2)^k\\ &\lt \varepsilon ^{1/3}(n+t+1-(k+1)(b+1))^k\\ &=\varepsilon ^{1/3}|V(\mathcal{F}_{t-b}(k,n';\;t-b))|^{k}. \end{align*}
Hence every vertex in
$H$
is
$\varepsilon ^{1/3}$
-good with respect to
$\mathcal{F}_{t-b}(k,n';\;t-b)$
. Note that
\begin{equation*} n'=n-kb\geq n-4k^2\sqrt {\varepsilon }n\gt (t-b)k/(1-\zeta ). \end{equation*}
By Lemma 15,
$H$
has a matching
$M'$
of size
$t-b$
. Therefore,
$M_1 \cup M'$
is a matching in
${{\mathcal F}}^{t+1}(k,n)$
of size
$t+1$
.
Hence, we may assume that for each
$x_i\in X\setminus X_1$
, there exists
$W^i \in \binom{[n] \setminus ([t]\setminus W_1)}{b}$
such that
$W^i$
is a vertex cover of
$\mathcal{F}_i$
. If
$W\;:\!=\;W^i=W^j$
for all
$x_i,x_j\in X\setminus X_1$
then
$([t] \setminus W_1)\cup W$
is a vertex cover of
$F_i$
for all
$i\in [t+1]$
; so
$\mathcal{F}^{t+1}(k,n)$
is a subgraph of
$\mathcal{F}_{t+1}(k,n;\;t)$
. Hence, we may assume that there exist
$i\ne j$
such that
$W^i \setminus W^j \ne \emptyset$
.
Now fix
$x \in X_1$
. Note that each edge in
$N_{\mathcal{F}^{t+1}(k,n)}(x)$
either intersects
$([t] \setminus W_1) \cup (W^i \cap W^j)$
or intersects both
$W^i \setminus W^j$
and
$W^j \setminus W^i$
. Recall that
$|W^i|=|W^j|=b$
. Let
$r\;:\!=\;|W^i \setminus W^j|=|W^j \setminus W^i|$
; then
$|([t]\setminus W_1) \cup (W^i \cap W^j)|= (t-b)+(b-r)=t-r$
. Hence,
\begin{align} |N_{\mathcal{F}^{t+1}(k,n)}(x)| \le \left ( \binom{n}{k} - \binom{n-(t-r)}{k} \right ) + \binom{n-t-r}{k-2} r^2. \end{align}
Note that
\begin{align*} \binom{n-(t-r)}{k} - \binom{n-t}{k} &\ge r \binom{n-t}{k-1}\\ & = \frac {(r-1)(n-t)}{k-1} \binom{n-t-1}{k-2} + \binom{n-t}{k-1}, \end{align*}
i.e.,
\begin{align} \binom{n-(t-r)}{k} \ge \frac {(r-1)(n-t)}{k-1} \binom{n-t-1}{k-2} + \binom{n-t}{k-1}+\binom{n-t}{k}. \end{align}
If
$r=1$
, then by (5), we have
\begin{align*} |N_{\mathcal{F}^{t+1}(k,n)}(x)| &\le \binom{n}{k} - \binom{n-(t-1)}{k}+ \binom{n-t-1}{k-2}\\ &=\binom{n}{k} - \binom{n-t}{k}-\binom{n-t-1}{k-1}\\ &\quad \lt \binom{n}{k} - \binom{n-t}{k} - \binom{n-t-k}{k-1}, \end{align*}
This leads to a contradiction as
$e\,(F_i)\gt \binom{n}{k}-\binom{n-t}{k} - \binom{n-t-k}{k-1} + 1$
for
$i\in [t+1]$
.
Now we may assume that
$r\geq 2$
. Combining (5) and (6), it follows that
\begin{align*} \begin{split} |N_{\mathcal{F}^{t+1}(k,n)}(x)| &\le \binom{n}{k} - \binom{n-t}{k} - \binom{n-t}{k-1} - \frac {(r-1)(n-t)}{k-1} \binom{n-t-1}{k-2} + \binom{n-t-1}{k-2} r^2 \\ &\lt \binom{n}{k} - \binom{n-t}{k} - \binom{n-t}{k-1} - \binom{n-t-1}{k-2} \quad (\text{as}\,\, n \gt 10kb \ge 10kr \,\, \text{and}\,\, t\le n/k)\\ &\lt \binom{n}{k} - \binom{n-t}{k} - \binom{n-t-1}{k-1} \\ &\lt \binom{n}{k} - \binom{n-t}{k} - \binom{n-t-k}{k-1}, \end{split} \end{align*}
contradicting that
$e\,(F_i)\gt \binom{n}{k}-\binom{n-t}{k} - \binom{n-t-k}{k-1} + 1$
for
$i\in [t+1]$
. This completes the proof.
5. Non-extremal case
To deal with the case when
$\mathcal{F}^{t+1}({k},n)$
is not
$\varepsilon$
-close to
$\mathcal{F}_{t+1}({k},n;\;t)$
, we need two lemmas proved by the present authors in [Reference Lu, Wang and Yu19], both with a slight variation but same proof.
The first is Lemma 4.2 in [Reference Lu, Wang and Yu19]. The only difference between that lemma and the statement below is that we replace the assumption ‘
$d_H(x_i)\gt \binom{n}{k}-\binom{n-t+1}{k}$
for
$i\in [t]$
’ in Lemma 4.2 of [Reference Lu, Wang and Yu19] by ‘
$d_H(x_i)\gt \binom{n}{k}-\binom{n-t+1}{k} - \rho n^k$
for
$i\in [t]$
’. We omit the proof as it is almost identical to the proof of Lemma 4.2 in [Reference Lu, Wang and Yu19].
Lemma 17.
Let
$n,k,t$
be integers and
$\zeta , \rho$
be real numbers, such that
$1/n\ll \rho \ll \zeta \ll 1/k$
and
$ n/(2k^3) \le t \le (1-\zeta )n/k$
. Let
$H$
be a
$(1,k)$
-partite
$(k+1)$
-graph with partition classes
$\{x_1, \ldots , x_{\lfloor n/k\rfloor } \}, [n]$
such that
$d_H(x_i)\gt \binom{n}{k}-\binom{n-t+1}{k} - \rho n^k$
for
$i\in [t]$
and
$d_H(x_i)=\binom{n}{k}$
for
$i=t+1, \ldots , \lfloor n/k\rfloor$
. Then for any
$c$
with
$0\lt c \ll \zeta$
, there exists a matching
$M$
in
$H$
such that
$|M|\le 2k c n$
and, for any balanced subset
$S\subseteq V(H)$
with
$|S|\le (k+1)c^{1.5} n/2$
,
$H[V(M)\cup S]$
has a perfect matching.
The second result we need is Lemma 6.4 in [Reference Lu, Wang and Yu19], with the assumption
$n/(3k^2) \le t$
replaced by
$n/(2k^3)\le t$
in the statement below. Again, the same proof in [Reference Lu, Wang and Yu19] works. Recall the definition of
$\mathcal{H}_t(k,n;\;t)$
.
Lemma 18.
Let
$k\ge 3$
be an integer,
$0 \lt c\ll \rho \ll \varepsilon \ll 1$
be real numbers,
$n\in k\mathbb{Z}$
sufficiently large, and let
$t$
be an integer with
$n/(2k^3) \le t \le (\frac {1}{2}+c)n/k$
. Let
$H$
be a
$(1,k)$
-partite
$(k+1)$
-graph with partition classes
$X,[n]$
such that
$k|X|=n$
. Let
$A_1$
and
$A_2$
be a partition of
$A$
such that
$|A_1|=t$
and
$|A_2|=n/k-t$
. Suppose that
$d_{H}(x)\gt \binom{n}{k}-\binom{n-t+1}{k}-\rho n^k$
for all
$x\in A_1$
and
$d_{H}(x)=\binom{n}{k}$
for all
$x\in A_2$
. If
$H$
is not
$\varepsilon$
-close to
$\mathcal{H}_t(k,n;\;t)$
, then there exists a spanning subgraph
$H'$
of
$H$
such that the following conditions hold:
-
(1) For all
$x\in V(H')$
, with at most
$n^{0.99}$
exceptions,
$d_{H'}(x)=(1\pm n^{-0.01})n^{0.2}$
. -
(2) For all
$x\in V(H')$
,
$d_{H'}(x)\lt 2 n^{0.2}$
. -
(3) For any two distinct
$x,y\in V(H')$
,
$d_{H'}(\{x,y\})\lt n^{0.19}$
.
Lemma 18 allows us to apply the following result attributed to Pippenger [Reference Pippenger and Spencer22] (see Theorem 4.7.1 in [Reference Alon and Spencer4]). An edge cover in a hypergraph
$H$
is a set of edges whose union is
$V(H)$
.
Theorem 19 (Pippenger). For every integer
$k\ge 2$
and reals
$r\ge 1$
and
$a\gt 0$
, there are
$\gamma =\gamma (k,r,a)\gt 0$
and
$d_0=d_0(k,r,a)$
such that for every
$n$
and
$D\ge d_0$
the following holds: Every
$k$
-uniform hypergraph
$H=(V,E)$
on a set
$V$
of
$n$
vertices in which all vertices have positive degrees and which satisfies the following conditions:
-
(1) for all vertices
$x\in V$
but at most
$\gamma n$
of them,
$d_H(x)=(1\pm \gamma )D$
; -
(2) for all
$x\in V$
,
$d_H(x)\lt r D$
; -
(3) for any two distinct
$x,y\in V$
,
$d_H(\{x,y\})\lt \gamma D$
;
contains an edge cover of at most
$(1+a)(n/k)$
edges.
Proof of Theorem 5.
By Corollary 13, we may assume that
$2k(t+1) \lt n\lt 2k^3 (t+1)$
. Let
$0 \lt \varepsilon \ll {1/k} \lt 1$
be sufficiently small and let
$t \gt t_0(k)=n/(2k)^3$
be sufficiently large. By Observation 6, it suffices to show
$\mathcal{F}^{t+1}(k,n)$
has a matching of size
$t+1$
. Applying Lemma 16 to
$\mathcal{F}^{t+1}(k,n)$
(by choosing
$\zeta \lt 1/3$
for instance), we may assume that
$\mathcal{F}^{t+1}(k,n)$
is not
$\varepsilon$
-close to
$\mathcal{F}_{t+1}(k,n;\;t)$
. That is,
$\mathcal{H}^{t+1}(k,n)$
is not
$\varepsilon /6$
-close to
$\mathcal{H}_{t+1}(k,n;\;t+1)$
by Observation 8.
Now we apply Lemma 17 to
$\mathcal{H}^{t+1}(k,n)$
with
$t\leq n/(2k)$
and sufficiently small
$\zeta$
. Thus there exists some constant
$0 \lt c \ll \varepsilon$
such that
$n-2k^2cn \ge (2k-2k^4c)(t+1)$
, and
$\mathcal{H}^{t+1}(k,n)$
contains an absorbing matching
$M_1$
with
$m_1\;:\!=\;|M_1|\leq 2kc n$
and for any balanced subset
$S$
of vertices with
$|S|\leq (k+1)c^{1.5} n$
,
$\mathcal{H}^{t+1}(k,n)[V(M_1)\cup S]$
has a perfect matching. Let
$H\;:\!=\;\mathcal{H}^{t+1}(k,n)-V(M_1)$
and
$n_1 \;:\!=\; n-km_1$
.
Next, we see that
$H$
is not
$(\varepsilon /12)$
-close to
$\mathcal{H}_{t+1}(k,n-km_1;t+1)$
. For, suppose otherwise. Then
\begin{align*} & |E(\mathcal{H}_{t+1}(k,n;\;t+1)) \setminus E(\mathcal{H}^{t+1}(k,n))| \\ &\le |E(\mathcal{H}_{t+1}(k,n-km_1;t+1)) \setminus E(H)| + |e \in E(\mathcal{H}_{t+1}(k,n;\;t+1)) \;:\; e \cap V(M_1) \neq \emptyset | \\ &\leq (\varepsilon /12) (n+n/k)^{k+1} + 2k(k+1)cn \cdot n^k \\ &\le \frac {1}{6}{\varepsilon (n+n/k)^{k+1}.} \end{align*}
This is a contradiction as
$\mathcal{H}^{t+1}(k,n)$
is not
$\varepsilon$
-close to
$\mathcal{H}_{t+1}(k,n;\;t+1)$
.
Let
$Q,[n]$
denote the partition classes of
$\mathcal{H}^{t+1}(k,n)$
, and let
$X$
consist of all vertices of
$Q$
contained in
$\mathcal{F}^{t+1}(k,n)$
. Let
$A_1=X\cap V(H)$
and
$A_2=(Q\setminus X)\cap V(H)$
. Then we have
$d_{H}(x)\gt \binom{n}{k}-\binom{n-t+1}{k}-\rho n^k$
for all
$x\in A_1$
and
$d_{H}(x)=\binom{n}{k}$
for all
$x\in A_2$
. Since
$n_1 =n-km_1\ge n - 2k^2cn \ge (2k-2 k^4 c)(t+1)$
, by Lemma 18
$H$
has a spanning subgraph
$H_1$
such that
-
(1) for all but at most
$n_1^{0.99}$
vertices
$x\in V(H_1)$
,
$d_{H_1}(x)=(1\pm n_1^{-0.01})n_1^{0.2}$
; -
(2) for all
$x\in V(H_1)$
,
$d_{H_1}(x)\lt 2 n_1^{0.2}$
; -
(3) for any two distinct
$x,y\in V(H_1)$
,
$d_{H_1}(\{x,y\})\lt n_1^{0.19}$
.
Hence by applying Lemma19 to
$H_1$
by choosing
$a$
with
$0 \lt a \ll c^{1.5}$
,
$H_1$
contains an edge cover of size at most
$(1+a)((n_1/k+n_1)/(k+1))$
. Thus, at most
$a(n_1/k+n_1)$
vertices are each covered by more than one edge in the cover. Hence, after removing at most
$a(n_1/k+n_1)$
edges from the edge cover, we obtain a matching
$M_2$
covering all but at most
$(k+1)a(n_1/k+n_1) \le 3kan_1 \le 3kan$
vertices.
Now we may choose a subset
$S$
of
$V(H)\setminus V(M_2)$
such that
$|V(H)\setminus (V(M_2)\cup S)|\lt k$
and if
$S\ne \emptyset$
then
$|S\cap [n]|={k|S\cap Q|}$
. Since
$|S| \le 3kan \le (k+1)c^{1.5}n$
,
$\mathcal{H}^{t+1}(k,n)[V(M_1)\cup S]$
has a perfect matching, say
$M_3$
. Thus,
$M_2\cup M_3$
is matching of
${\mathcal H}^{t+1}(k,n)$
covering all but at most
$k$
vertices, and, hence, has size
$\lfloor n/k\rfloor$
. Therefore, by Lemma 7,
$\mathcal{F}^{t+1}(k,n)$
has a matching of size
$t+1$
.
6. Conclusion
We proved stability results for rainbow matchings for families of non-uniform hypergraphs. In the uniform case, our result generalizes the result of Frankl and Kupavskii [Reference Frankl and Kupavskii12] on a conjecture which we now describe.
Let
$n,k,s$
be positive integers. For
$i\in [k]$
, define
\begin{equation*}{\mathcal A}_i^{(k)}(n,s)\;:\!=\;\left \{A\in \binom{[n]}{k} \;: \left | A\cap [(s+1)i-1] \right | \ge i\right \}.\end{equation*}
Note that
$|{\mathcal A}_i^{(k)}(n,s)|\le \binom{n}{k} -\sum _{j=0}^{i-1}\binom{(s+1)i-1}{j}$
. It is worth pointing out that
${\mathcal A}_k^{(k)}(n,t-1)$
and
${\mathcal A}_1^{(k)}(n,t-1)$
, respectively, correspond to the extremal constructions (a) an (b) of the Erdős Matching Conjecture. Let
${\mathcal H}^{(k)}(n,s)$
denote the family consisting of the set
$\{s+1,\ldots ,s+k\}$
and all
$k$
-sets that either intersect
$[s-1]$
, or contain
$s$
and intersect
$[s+k]\setminus [s]$
, i.e.
\begin{equation*}\left \{ e \in \binom{[n]}{k} \;:\; \min (e) \lt s \text{ or } \min (e) = s \text{ and } |e \cap [s+k]| \ge 2 \right \} \cup \{ [s+k] \setminus [s] \}\end{equation*}
Note that
${\mathcal H}^{(k)}(n,s)$
is a slight variation of
${\mathcal A}_1^{(k)}(n,s)$
,
$\nu ({\mathcal H}^{(k)}(n,s))=s$
, and
$|{\mathcal H}^{(k)}(n,s)|=\binom{n}{k}-\binom{n-s}{k}-\binom{n-s-k}{k-1}+1$
.
Frankl and Kupavskii [Reference Frankl and Kupavskii12] proposed the following conjecture.
Conjecture 20.
Let
$n,k,t$
be positive integers such that
$n\gt kt$
. Let
$F$
be a
$k$
-graph of order
$n$
. If
$\nu (F)\leq t$
, then either
$\tau (F)\le t$
or
\begin{equation*} e(F)\leq \max \{e({\mathcal H}^{(k)}(n,t)), e({\mathcal A}_2^{(k)}(n,t)), \ldots , e({\mathcal A}_k^{(k)}(n,t)) \}. \end{equation*}
The case
$t = 1$
of Conjecture 20 is the Hilton-Milner theorem, which is proved in [Reference Hilton and Milner14]. Frankl and Kupavskii [Reference Frankl and Kupavskii12] confirmed the conjecture for
$n\geq (2+o(1))kt$
. Theorem1 extends this result to rainbow matchings.
We think the following could be true:
Conjecture 21.
Let
$n,k,t$
be three positive integers such that
$n \gt kt$
. Let
${{\mathcal F}}=\{F_1,\ldots , F_{t+1}\}$
be a family of
$k$
-graphs, such that, for
$1\leq i\leq t+1$
,
\begin{equation*} |F_i| \gt \max \{e({\mathcal H}^{(k)}(n,t)), e({\mathcal A}_2^{(k)}(n,t)), \ldots , e({\mathcal A}_k^{(k)}(n,t)) \}. \end{equation*}
Then
${\mathcal F}$
admits a rainbow matching of size
$t+1$
or there exists
$W\in \binom{[n]}{t}$
such that for all
$i\in [t+1]$
,
$W$
is a vertex cover of
$F_i$
.
