Clique-factors in graphs with sublinear $\ell$-independence number

Erdős and Sós [14] initiated the study of a variant of Turán problem which excludes all graphs with large independence number. More generally, for an integer $\ell\geq 2$ and a graph $G$, the $\ell$-independence number of $G$, denoted by $\alpha_{\ell}(G)$, is the maximum size of a $K_{\ell}$-free subset of vertices. Given integers $n,r$ and a function $f(n)$, we use $\textbf{RT}_{\ell}(n,K_{r},f(n))$ to denote the maximum number of edges of an $n$-vertex $K_{r}$-free graph $G$ with $\alpha_{\ell}(G)\leq f(n)$. In particular, the Ramsey–Turán density of $K_{r}$ is defined as $\varrho_{\ell}(r):=\lim\limits_{\alpha\to 0}\lim\limits_{n\to\infty}\frac{\textbf{RT}_{\ell}(n,K_{r},\alpha n)}{\binom{n}{2}}$. Szemerédi [38] first showed that $\varrho_{2}(4)\leq\frac{1}{4}$. This turned out to be sharp as Bollobás and Erdős [7] provided a matching lower bound using an ingenious geometric construction. There are some recent exciting developments in this area [3, 4, 16, 24, 31, 33]. For further information on Ramsey–Turán theory the reader is referred to a comprehensive survey [37] by Simonovits and Sós.

Note that the extremal example that achieves the optimality of the bound on $\delta(G)$ in the Hajnal–Szemerédi theorem also has large independence number [6], which makes it far from being typical. Following the spirit of the Ramsey–Turán theory, a natural question on the Hajnal–Szemerédi theorem is whether the minimum degree condition can be weakened when the host graph has sublinear independence number. The following Ramsey–Turán type problem was proposed by Balogh, Molla and Sharifzadeh [6].

Balogh, Molla and Sharifzadeh [6] studied $K_{3}$-factors and showed that if the independence number of an $n$-vertex graph $G$ is $o(n)$ and $\delta(G)\geq\frac{n}{2}+\varepsilon n$ for any $\varepsilon>0$, then $G$ contains a triangle factor. Recently Knierim and Su [26] resolved the case $r\geq 4$ by determining the asymptotically tight minimum degree bound $(1-\frac{2}{r})n+o(n)$.

The following problem was proposed by Nenadov and Pehova [35].

Nenadov and Pehova [35] also provided upper and lower bounds on the minimum degree condition. In particular, they solved Problem 1.2 for $r=\ell+1$ and proved that $n/2+o(n)$ is the correct minimum degree threshold. Knierim and Su [26] reiterated Problem 1.2 in their paper and proposed a minimum degree condition as follows.

Very recently, Chang, Han, Kim, Wang and Yang [8] determines the asymptotically optimal minimum degree condition for $\ell\geq\frac{3}{4}r$, which solves Problem 1.2 for this range, and indeed provides a negative answer to Problem 1.3.

Based on this result, Problem 1.3 should be revised as follows.

By the aformentioned results, Problem 1.5 is solved for $\ell=2$ [26] and for $\ell\geq 3r/4$ [8], both done by quite involved proofs. It seems to us that a complete resolution of Problem 1.5 is a quite challenging task.

The purpose of this paper is to extend the discussion on the problem to sublinear $\ell$-independence numbers. We first state a simplified version of our main result which says that the answer to Problem 1.5 is yes if we assume a slightly stronger assumption $\alpha_{\ell}(G)\leq n^{1-o(1)}$.

To state our main result, we define more general versions of the Ramsey–Turán densities as follows.

By definition trivially it holds that $\varrho^{*}_{\ell}(r,f)\leq\varrho_{\ell}(r,f)$. It is proved in [8] that $\varrho^{*}_{\ell}(r,f)=\varrho_{\ell}(r,f)$ if there exists $c\in(0,1)$ such that $xf(n)\leq f(x^{1/c}n)$ for every $x\in(0,1)$ and $n\in\mathbb{N}$.

The full version of our result is stated as follows.

In fact, Theorem 1.8 implies Theorem 1.6 by the observation that $\varrho^{*}_{\ell}(r-1,f)\leq\varrho^{*}_{\ell}(r-1)=\varrho_{\ell}(r-1)$ holds for any function $f(n)=n^{c}$ with $c\in(0,1)$. We shall supply lower bound constructions (see next subsection) which show that the minimum degree condition in Theorem 1.8 is asymptotically best possible for $\alpha_{\ell}(G)\in(n^{1-\gamma},n^{1-\varepsilon})$, for any $\gamma\in(0,\frac{\ell-1}{\ell^{2}+2\ell})$ and any $\varepsilon>0$. This can be seen as a stepping stone towards a full understanding of the Ramsey–Turán tiling thresholds for cliques where $\alpha_{\ell}(G)\in[2,o(n)]$.

Here we also provide some concrete thresholds that we could spell out from Theorem 1.8. Recall that Nenadov and Pehova [35] solved Problem 1.2 for $r=\ell+1$ whilst Knierim and Su [26] solved the case $\ell=2,r\geq 4$. Now we consider the first open case $r=\ell+2,\ell\geq 3$. Note that $\varrho^{*}_{\ell}(\ell+1,f)=\varrho_{\ell}(\ell+1)=0$. Then Theorem 1.4 says that for $\ell\geq 6$ (that is, $\ell\geq\frac{3}{4}(\ell+2)$) and $\alpha_{\ell}(G)=o(n)$, $\frac{n}{2}+o(n)$ is the minimum degree threshold forcing a $K_{\ell+2}$-factor. For the remaining cases $\ell\in\{3,4,5\}$, Theorem 1.8 implies that the minimum degree threshold is $\frac{n}{2}+o(n)$ under a stronger condition $\alpha_{\ell}(G)\leq n^{1-\varepsilon}$ for any fixed $\varepsilon>0$.

Our proof of Theorem 1.8 uses the absorption method and the regularity method. In particular, we use dependent random choice for embedding cliques in regular tuples with sublinear independence number, which is closely related to the Ramsey–Turán problem.

We note that both terms in the minimum degree condition in Theorem 1.8 are asymptotically best possible. First, we show that the first term cannot be weakened when $\alpha_{\ell}(G)\in(n^{1-\gamma},o(n))$ for some constant $\gamma$ as follows.

Indeed, Proposition 1.9 also gives, in the setting that $\alpha_{\ell}(G)\leq n^{1-o(1)}$, a lower bound construction for the minimum degree condition forcing an almost $K_{r}$-tiling that leaves a constant number of vertices uncovered. More results on almost graph tilings can be found in a recent comprehensive paper [19].

The second term $\frac{1}{2-\varrho^{*}_{\ell}(r-1,f)}$ is also asymptotically tight, which is given by a cover threshold construction as follows.
Cover Threshold. To have a $K_{r}$-factor in $G$, a naive necessary condition is that every vertex $v\in V(G)$ is covered by a copy of $K_{r}$ in $G$. The cover threshold has been first discussed in [20] and appeared in a few different contexts [8, 9, 12].

Now we give a construction that shows the optimality of the term $\frac{1}{2-\varrho^{*}_{\ell}(r-1,f)}$ for the function $f(n)$ as in Theorem 1.8. A similar construction can be found in [8]. Given integers $r,\ell$ and constants $\varepsilon,\alpha,x\in(0,1)$, we construct (for large $n\in r\mathbb{N}$) an $n$-vertex graph $G$ by

1. $(i)$

   first fixing a vertex $v$ such that $N(v)=xn$ and $G[N(v)]=:G^{\prime}$ is a $K_{r-1}$-free subgraph with $\delta(G^{\prime})\geq\varrho^{*}_{\ell}(r-1,f)xn-\varepsilon n$ and $\alpha_{\ell}(G^{\prime})\leq f(\alpha xn)$;

2. $(ii)$

   and then adding a clique of size $n-xn-1$ that is complete to $N(v)$.

There exists no copy of $K_{r}$ covering $v$ and thus $G$ contains no $K_{r}$-factor; moreover, by choosing $x=\tfrac{1}{2-\varrho^{*}_{\ell}(r-1,f)}$, we obtain $\delta(G)\geq\frac{1}{2-\varrho^{*}_{\ell}(r-1,f)}n-\varepsilon n$ and $\alpha_{\ell}(G)=\alpha_{\ell}(G^{\prime})\leq f(\alpha n)$.

When we write $\alpha\ll\beta\ll\gamma$, we always mean that $\alpha,\beta,\gamma$ are constants in $(0,1)$, and $\beta\ll\gamma$ means that there exists $\beta_{0}=\beta_{0}(\gamma)$ such that the subsequent arguments hold for all $0<\beta\leq\beta_{0}$. Hierarchies of other lengths are defined analogously. In the remaining proofs, we always take $\lambda=1/\lfloor\frac{r}{\ell}+1\rfloor$ and $f(n)\leq n^{1-\omega(n)\log^{-\lambda}n}$ unless otherwise stated.

Following typical absorption strategies, our main work is to establish an absorbing set (see Lemma 2.2) and find an almost-perfect tiling (see Lemma 2.3). We first introduce the following notions of absorbers and absorbing sets from [35].

Now we are ready to state our first crucial lemma, whose proof can be found in Section 4.

Our second crucial lemma is on almost $K_{r}$-factor as follows, whose proof will be given in Section 3.

Now we are ready to prove Theorem 1.8 using Lemma 2.2 and Lemma 2.3.

Given a graph $G$ and a pair $(X,Y)$ of vertex-disjoint subsets in $V(G)$, the density of $(X,Y)$ is defined as

For constants $\varepsilon,d>0$, we say that $(X,Y)$ is an $\varepsilon$-regular pair with density at least $d$ (or $(X,Y)$ is $(\varepsilon,d)$-regular) if $d(X,Y)\geq d$ and for all $X^{\prime}\subseteq X$, $Y^{\prime}\subseteq Y$ with $|X^{\prime}|\geq\varepsilon|X|$, $|Y^{\prime}|\geq\varepsilon|Y|$, we have

Moreover, a pair $(X,Y)$ is called $(\varepsilon,d)$-$super$-$regular$ if $(X,Y)$ is $(\varepsilon,d)$-regular, $d_{Y}(x)\geq d|Y|$ for all $x\in X$ and $d_{X}(y)\geq d|X|$ for all $y\in Y$. The following fact is an easy consequence of the definition of regularity.

Given a family of vertex-disjoint sets in $V(G)$ which are pairwise $\varepsilon$-regular, we can find in each set a large subset such that every pair of resulting subsets is super-regular.

We now state a degree form of the regularity lemma (see [27, Theorem 1.10]).

A widely-used auxiliary graph accompanied with the regular partition is the reduced graph. The $d$-reduced graph $R_{d}$ of $\mathcal{P}$ is a graph defined on the vertex set $\{V_{1},\ldots,V_{k}\}$ such that $V_{i}$ is connected to $V_{j}$ by an edge if $(V_{i},V_{j})$ has density at least $d$ in $G^{\prime}$. So if $V_{i}$ is not connected to $V_{j}$, then $(V_{i},V_{j})$ has density $0$ by property $(e)$ above. To ease the notation, we use $d_{R}(V_{i})$ to denote the degree of $V_{i}$ in $R_{d}$ for each $i\in[k]$. Note that $R_{d}$ also can be regarded as a weighted graph in which the weight for each edge $V_{i}V_{j}$, denoted by $d_{ij}$ for simplicity, is exactly the density of the pair $(V_{i},V_{j})$ in $G^{\prime}$.

To find an almost perfect $K_{r}$-tiling, we shall also make use of the following result of Komlós [28] on graph tilings. Given a graph $H$ on $r$ vertices, the critical chromatic number of $H$ is defined as $\chi_{cr}(H)=\frac{(k-1)r}{r-\sigma},$ where $k=\chi(H)$ and $\sigma=\sigma(H)$ denotes the smallest size of a color class over all $k$-colorings of $H$.

Based on this result, we will first apply the regularity lemma to $G$ to get a reduced graph $R:=R_{d}$ for a constant $d>0$, and then apply Theorem 3.5 to get an $H$-tiling of $R$ covering almost all vertices for a suitably-chosen auxiliary graph $H$. To get an almost $K_{r}$-tiling of $G$ from this almost $H$-tiling of $R$, we will use the following lemma which says if we can find a $K_{q}$ in a copy of $H$ in $R$, then we can find a $K_{pq}$ in $G$ under certain conditions on $\alpha_{p}(G)$. Its proof follows from that of Claim 6.1 in [5], where a similar assumption on $\alpha(G)$ (instead of $\alpha_{p}(G)$) is used. For completeness we include a proof of Lemma 3.6 in the appendix.