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Lihua Feng

Publications and source records attributed to Lihua Feng.

At least 19 recordsLinked to original sources
arXiv2026

A complete solution to the Tokushige measure conjecture and its stability

We resolve three conjectures proposed by Tokushige in 2013 about cross $t$-intersecting families of subsets and integer sequences. For $0 (1-\varepsilon)^2(p_1p_2)^t$, then there exists $T\in\binom{[n]}{t}$ such that $\mu_{p_i}(\mathcal F_i\mathbin{\triangle}\mathcal S_T)<C\varepsilon$ for $i=1,2$, where $C$ depends only on $t,p_1,p_2$. This improves Tokushige's conjectured $C\sqrt{\varepsilon}$ estimate to $C\varepsilon$. For integer sequences, we prove that if every sequence in $\mathcal H_1\subseteq[m]^n$ agrees with every sequence in $\mathcal H_2\subseteq[m]^n$ in at least $t$ coordinates, then $|\mathcal H_1||\mathcal H_2|\leq m^{2(n-t)}$ for all $n\geq t\geq1$ and $m\geq t+1$. We further obtain a more general result in which a separate agreement requirement is imposed for each possible value. This extends a theorem of Frankl and Kupavskii and recovers their earlier cross intersection--union product theorem.

math.CO↗
arXiv2026

A Spectral Hilton--Milner--Frankl Theorem for $t$-Intersecting Families

Keevash, Lenz, and Mubayi proved a spectral Erd\H{o}s--Ko--Rado theorem, showing that, for sufficiently large $n$, the complete $t$-star uniquely maximizes the adjacency-tensor spectral radius among all $t$-intersecting $k$-uniform families. In this paper, we establish a spectral Hilton--Milner--Frankl theorem for nontrivial $t$-intersecting families in the explicit range $1\le t\le k-2$ and $n\ge 100\cdot 2^k k^7$. More precisely, we prove that, for every nontrivial $t$-intersecting $k$-uniform family $\mathcal F$, the spectral radius satisfies \[ \rho(\mathcal F)\le \max\{\rho(\mathcal H_{n,k,t}),\rho(\mathcal A_{n,k,t})\}, \] where $\mathcal H_{n,k,t}$ and $\mathcal A_{n,k,t}$ are the two extremal families appearing in the classical Hilton--Milner--Frankl theorem. Moreover, equality holds only for the extremal candidates attaining the maximum, up to isomorphism. We further compare the two candidates asymptotically. For each fixed $t$, the unique real solution $x=x_t$ of \[ (t+2)^{x-t-1}(t+1)^{t+1}=(x-t+1)^{x-1} \] determines, as $k$ varies, which of $\mathcal H_{n,k,t}$ and $\mathcal A_{n,k,t}$ has the larger asymptotic spectral radius.

math.CO↗
arXiv2026

Rigidity and stability for biased cross-intersecting families

Let $\mathbf p=(p_1,\ldots,p_n)$ and $\mathbf q=(q_1,\ldots,q_n)$ belong to $(0,1/2]^n$, and let $\mu_{\mathbf p}$ and $\mu_{\mathbf q}$ be the associated measures on $2^{[n]}$. Suppose that $p_1q_1=\max_{i\in[n]}p_iq_i$. We prove that every pair of cross-intersecting families $\mathcal A,\mathcal B\subseteq2^{[n]}$ satisfies the sharp inequality $\mu_{\mathbf p}(\mathcal A)\mu_{\mathbf q}(\mathcal B)\leq p_1q_1$. This confirms a conjecture of Suda, Tanaka and Tokushige [Math. Program. 166 (2017) 113--130]. We also determine all equality cases. When $p_1q_1<1/4$, equality is attained only when both families consist of all subsets containing the same product-maximizing coordinate. At the endpoint $p_1q_1=1/4$, we identify precisely the additional extremal pairs, which are induced by half-sized increasing families on the coordinates satisfying $p_i=q_i=1/2$. We further resolve the remaining conjecture from the same paper by proving a dimension-free stability theorem. Assume that the first coordinate has maximum probability under both measures and that $p_1,q_1<1/2$. If $\mu_{\mathbf p}(\mathcal A)\mu_{\mathbf q}(\mathcal B)\geq(1-\varepsilon)p_1q_1$, then there exists a coordinate $j$ such that both $\mathcal A$ and $\mathcal B$ are within $c(p_1,q_1)\varepsilon$, in their respective measures, of the family of all subsets containing $j$. This improves the conjectured $O(\sqrt{\varepsilon})$ bound to a linear one. The main new ingredient in the sharp measure theorem is a log-odds interpolation combined with induction on coordinate sections, while stability follows from a semidefinite estimate and a one-coordinate approximation theorem.

math.CO↗
arXiv2026

The binomial norm of intersecting-union families

In a 2021 survey on Katona's circle method, Frankl conjectured that every family $\mathcal{F}\subseteq 2^{[n]}$ in which any two members intersect and no two members cover $[n]$ satisfies the sharp binomial norm bound $ \lVert\mathcal F\rVert_n :=\sum_{F\in\mathcal F}\binom{n}{|F|}^{-1} \leq \frac{n+1}{6}. $ This improves the earlier estimate $\frac{n}{4}$ obtained by the circle method. In this paper, we prove Frankl's conjecture and determine all extremal families. Our proof develops a continuous $p$-biased measure approach in place of the circle method. The intersection and union conditions lead to a sharp estimate for $ \mu_p(\mathcal F)+\mu_{1-p}(\mathcal F). $ Integrating this estimate over $p$ converts it directly into the desired binomial norm bound and recovers the optimal coefficient $\frac{1}{6}$. This continuous averaging is the key new ingredient of the proof and also yields the characterization of all extremal families.

math.CO↗
arXiv2026

On a conjecture regarding the product version of the Hilton-Milner theorem

Recently, Frankl and Wang considered a product version of the classical Hilton-Milner theorem. They conjectured that, if $\mathcal{F} \subset \binom{[n]}{k}$ and $\mathcal{G} \subset \binom{[n]}{\ell}$ are non-trivial cross-intersecting families with $n \geq 2k > 2\ell \geq 4$, the maximum of $|\mathcal{F}||\mathcal{G}|$ is attained by the natural Hilton-Milner-type configurations. In this paper, we present two main results concerning this conjecture. Firstly, we show that the conjecture does not hold in general. By introducing a two-center construction, we prove that for every fixed integer $\ell \geq 3$ and all sufficiently large $k$, the conjecture is false in a linear range $2k+1 \leq n \leq (c_\ell - \epsilon)k$ for any $0 < \epsilon < c_\ell - 2$, where $c_\ell > 2$ is an explicit constant. Secondly, we prove that the conjecture holds when $n > 100\ell k^2$ and $3 \leq \ell < k$, and we completely characterize the extremal families. Our proofs rely on the size of minimal covers and analyzing the structural properties of $2$-cover graphs.

math.CO↗
arXiv2026

Random partition for Tokushige's $r$-wise intersecting conjecture

Let $r\ge 3$ and let $1>p_1\ge p_2\ge\cdots\ge p_n>0$. Let $\mu_{\mathbf p}$ denote the product measure on $2^{[n]}$ where each coordinate $i$ is included independently with probability $p_i$. A family $\mathcal A\subseteq 2^{[n]}$ is $r$-wise intersecting if $A_1\cap\cdots\cap A_r\neq\emptyset$ for all $A_1,\ldots,A_r\in\mathcal A$. In 2022, Tokushige proved that if $p_2<\frac{r-1}{r}$, then every $r$-wise intersecting family $\mathcal{A}\subseteq 2^{[n]}$ satisfies $\mu_{\mathbf p}(\mathcal{A})\le p_1$, with equality only for stars centred at coordinates of maximum probability. He conjectured that the hypothesis $p_2<\frac{r-1}{r}$ can be replaced by $p_{r+1}<\frac{r-1}{r}$. In this paper, we prove this conjecture in full. The key novelty is the introduction of a new random partition method, which reduces the problem to at most $r$ coordinates and solves it exactly, thereby fully covering all cases with multiple supercritical coordinates.

math.CO↗
arXiv2026

The Suda-Tanaka-Tokushige conjecture for $\mathbf{p}$-biased intersecting families

In 2017, Suda, Tanaka and Tokushige conjectured that if $1>p_1\ge\cdots\ge p_n>0$ with $p_3\le \frac{1}{2}$, then every intersecting family $\mathcal A\subseteq 2^{[n]}$ satisfies $\mu_{\mathbf{p}}(\mathcal A)\le p_1$, where $\mu_{\mathbf{p}}$ is the non-uniform product measure defined by $\mu_{\mathbf{p}}(\mathcal{A})=\sum_{A\in\mathcal{A}} \prod_{i\in A} p_i \prod_{j\in [n]\setminus A}(1-p_j)$. In addition, if $p_1 > p_3$ or $p_1 < \frac{1}{2}$, then equality holds if and only if $\mathcal{A}$ is a star centered at some $i \in [n]$ with $p_i = p_1$. In this paper, we prove this conjecture in the following stronger $t$-intersecting form: for any $t\ge 1$, if $p_{t+2}\le \frac{1}{t+1}$, then every $t$-intersecting family $\mathcal{A} \subseteq 2^{[n]}$ satisfies $\mu_{\mathbf{p}}(\mathcal A)\le \prod_{i=1}^t p_i$. Moreover, when $p_{t+2}<\frac{1}{t+1}$, equality holds if and only if $\mathcal{A}=\{A\subseteq [n]: T\subseteq A\}$ for some $T\in \binom{[n]}{t}$ with $\prod_{i\in T} p_i=\prod_{i=1}^t p_i$. Our result unifies and generalizes the classical theorems of Fishburn-Frankl-Freed-Lagarias-Odlyzko and Friedgut.

math.CO↗
arXiv2026

Improved bound on symmetric differences of intersecting families

For a family $\mathcal{F}$, it is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in\mathcal{F}$. We use $\mathcal{SD}(\mathcal{F}) = \{F \triangle G : F, G \in \mathcal{F}\}$ to denote the family of symmetric differences of $\mathcal{F}$. In 2023, Frankl, Kiselev and Kupavskii conjectured that for any intersecting family $\mathcal{F} \subseteq \binom{[n]}{k}$ with $n > 10k$, the inequality $|\mathcal{SD}(\mathcal{F})| \le \sum_{\ell=0}^{k-1} \binom{n-1}{2\ell}$ holds. They further observed that a proof for the range $n>3k^2$ could likely be obtained via arguments similar to those in their earlier work, though no detailed derivation was given. In this paper, we establish the conjecture under the conditions $n\ge 100k\ln k$ and $k\ge 50$. We also determine the extremal families, which are precisely a certain class of stars. A concentration inequality plays a central role in the proof.

math.CO↗
arXiv2026

Bollob\'{a}s-type inequalities for subspaces via weight invariance

Let $V$ be an $n$-dimension real vector space with a direct sum decomposition $V = V_1 \oplus \cdots \oplus V_r$. Let $\mathcal{P} = \{(A_i, B_i) : i \in [m]\}$ be a skew Bollob\'as system of subspaces of $V$ such that each $i\in [m]$, $ A_i = \bigoplus_{k=1}^r (A_i \cap V_k)$ and $ B_i = \bigoplus_{k=1}^r (B_i \cap V_k)$. We prove that $$\sum_{i=1}^{m} \prod_{k=1}^{r} \left[ \binom{a_{i,k} + b_{i,k}}{a_{i,k}} (1 + a_{i,k} + b_{i,k})^{-1} \right] \leq 1,$$ where $a_{i,k} = \dim(A_i \cap V_k)$ and $b_{i,k} = \dim(B_i \cap V_k)$. This extends a recent result of Yue from set systems to finite dimensional subspaces. We then consider Tuza's theorem on weak Bollob\'as system for $d$-tuples. We give an alternative proof of the original set version of Tuza, and also establish its vector space analogue. Precisely, let $\mathcal{P} = \{(A_i^{(1)}, \ldots, A_i^{(d)}) : i \in [m]\}$ be a skew Bollob\'as system of $d$-tuples of subspaces of finite dimensional space $V$ with $a^{(\ell)}_i=\dim (A_i^{(\ell)})$. Then, for any positive real numbers $p_1, \ldots, p_d$ satisfying $p_1 + \cdots + p_d = 1$, we prove that $ \sum_{i=1}^{m} \prod_{\ell=1}^{d} p_{\ell}^{a_i^{(\ell)}} \leq 1. $

math.CO↗
arXiv2026

Subspace variations of the weighted skew Bollob\'as theorem

Let $V$ be a finite-dimensional real vector space. A collection $\mathcal{P} = \{(A_i,B_i)\}_{i=1}^m$ of pairs of subspaces of $V$ is called a skew Bollob\'as system if $\dim(A_i\cap B_i)=0$ for each $i\in [m]$ and $\dim(A_i\cap B_j)>0$ for all $1\leq i<j \leq m$. Assume that $V = V^{(1)}\oplus \cdots \oplus V^{(r)}$ and $\mathcal{P}= \{(A_i,B_i)\}_{i=1}^m$ is a skew Bollob\'as system of subspaces of $V$ satisfying $ A_i = \bigoplus_{k=1}^r (A_i \cap V^{(k)})$ and $ B_i = \bigoplus_{k=1}^r (B_i \cap V^{(k)})$ for each $i\in [m]$. Denote $a_{i,k} = \dim(A_i \cap V^{(k)})$ and $b_{i,k} = \dim(B_i \cap V^{(k)})$. Suppose that $a_{1,k} \le \cdots \le a_{m,k}$ and $b_{1,k} \ge \cdots \ge b_{m,k}$ for each $k\in [r]$. Using the exterior algebraic method developed by Lov\'{a}sz and Scott--Wilmer, we prove that $$ \sum_{i=1}^{m} \frac{1}{\prod_{k=1}^{r} \binom{a_{i,k}+b_{i,k}}{a_{i,k}}} \le 1 . $$ This generalizes the results of Alon (JCTA, 1985) and Scott--Wilmer (JLMS, 2021) to multipart weighted setting. Secondly, we solve a conjecture of Heged\"us (AJC, 2015) concerning projective subspaces, showing that any skew Bollob\'as system of projective subspaces in an $n$-dimensional projective space contains at most $2^{n+1} - 2$ pairs. Thirdly, we prove that if $\mathcal{P}= \{(A_i,B_i)\}_{i=1}^m$ is a skew Bollob\'as system of subspaces of $V$ with $a_i=\dim (A_i)$ and $b_i=\dim (B_i)$, then $$ \sum_{i=1}^m \frac{1}{(a_i+ b_i+1)\binom{a_i+b_i}{a_i}} \le 1. $$ This gives an extension to the subspace setting of the results of Heged\"us--Frankl (EUJC, 2024) and Yue (DM, 2026). Finally, we extend the above inequality to systems of $d$-tuples of subspaces, giving a unified bound that implies the corresponding results for $d$-tuples of subsets.

math.CO↗
arXiv2026

Coupling Lattice Distortion and Cation Disorder to Control Li-ion Transport in Cation-Disordered Rocksalt Oxides

Cation-disordered solids offer a rich chemical landscape where local coordination, lattice responses, and configurational disorder collectively, yet often implicitly, govern ion transport. In cation-disordered rocksalt oxides, Li+ diffusion has conventionally been rationalized by the static 0-transition-metal (0-TM) percolation rule, which assumes an ideal, passive lattice and thus fails to capture experimentally accessible capacities. Here, we show that lattice distortion is an essential, previously overlooked degree of freedom that actively reshapes Li+ percolation networks. By developing a lattice-responsive framework combining Monte Carlo sampling of cation configurations with machine-learning-accelerated molecular dynamics, we quantitatively predict Li+ percolation and electrochemical capacities within 5% of experiment. Our results reveal a causal coupling between lattice distortion and cation short-range order: enhanced local distortions precede and suppress short-range ordering, activating Li+ migration through nominally inaccessible 1-TM channels, fundamentally extending percolation beyond the 0-TM paradigm. Guided by this, we design and synthesize a high-entropy oxide, Li1.2Mn0.2Ti0.2V0.2Mo0.2O2, which exhibits enhanced distortion and achieves a 71.9% Li+ percolation network, surpassing 65.8% in Li1.2Mn0.4Ti0.4O2, delivering 256.3 mAh/g capacity, closely matching our prediction of 255.1 mAh/g. These findings establish lattice distortion as an active control parameter for ion transport, revising percolation concepts and offering a general design principle beyond metal-ion cathodes.

cond-mat.mtrl-sci↗
arXiv2025

The number of rooted spanning forests of bicirculant graphs

A bi-Cayley graph over the cyclic group $(\mathbb{Z}_n, +)$ is called a bicirculant graph. Let $\Gamma=BC(\mathbb{Z}_n; R,T,S)$ be a bicirculant graph with $R=-R\subseteq \mathbb{Z}_n\setminus \{0\}$ and $T={-}T\subseteq \mathbb{Z}_n\setminus \{0\}$ and $S\subseteq \mathbb{Z}_n$. In this paper, using Chebyshev polynomials, we obtain a closed formula for the number of rooted spanning forests of $\Gamma$. Moreover, we investigate some arithmetic properties of the number of rooted spanning forests of $\Gamma$, and find its asymptotic behaviour as $n$ tends infinity.

math.CO↗
arXiv2025

The product measures of cross $t$-intersecting families

We investigate the product measures of intersection problems in extremal combinatorics. Invoking a recent result of He--Li--Wu--Zhang, we prove that for any $ n \geq t \geq 3$ and $ p_1, p_2 \in (0, \frac{1}{t+1})$, if $ \mathcal{F}_1, \mathcal{F}_2 \subseteq 2^{[n]}$ are cross $ t$-intersecting families, then $\mu_{p_1}(\mathcal{F}_1)\mu_{p_2}(\mathcal{F}_2)\le (p_1p_2)^t$. Secondly, we study the intersection problems for integer sequences by proving that if $\mathcal{H}_1, \mathcal{H}_2 \subseteq [m]^{n}$ are cross $t$-intersecting with $ m > t+1$, then $|\mathcal{H}_1|| \mathcal{H}_2|\leq (m^{n-t})^2$. These results confirm two classical conjectures of Tokushige. As an application, we strengthen a recent theorem of Frankl--Kupavskii, generalizing the well-known IU-Theorem. Finally, we show that if $ p \geq \frac{1}{2}$ and $ \mathcal{F}_1, \mathcal{F}_2 \subseteq 2^{[n]}$ are cross $t$-intersecting families, then $\min \left\{\mu_{p}(\mathcal{F}_1),\mu_{p}(\mathcal{F}_2)\right\} \leq \mu_{p}(\mathcal{K}(n,t))$, where $\mathcal{K}(n,t)$ denotes the Katona family. This recovers an old result of Ahlswede--Katona.

math.CO↗
arXiv2025

Two results on set families: sturdiness and intersection

This paper resolves two open problems in extremal set theory. For a family $\mathcal{F} \subseteq 2^{[n]}$ and $i, j\in [n]$, we denote $\mathcal{F} (i,\bar{j})=\{F\backslash\{i\}: F\in \mathcal{F}, F\cap\{i,j\}=\{i\}\}$. The sturdiness $\beta (\mathcal{F})$ is defined as the minimum $|\mathcal{F} (i,\bar{j})|$ over all $i\neq j$. A family $\mathcal{F}$ is called an IU-family if it satisfies the intersection constraint: $F\cap F'\neq \emptyset $ for all $F,F'\in \mathcal{F}$, as well as the union constraint: $F\cup F' \neq [n]$ for all $F,F'\in \mathcal{F}$. The well-known IU-Theorem states that every IU-family $\mathcal{F}\subseteq 2^{[n]}$ has size at most $ 2^{n-2}$. In this paper, we prove that if $\mathcal{F}\subseteq 2^{[n]}$ is an IU-family, then $\beta (\mathcal{F})\le 2^{n-4}$. This confirms a recent conjecture proposed by Frankl and Wang. As the second result, we establish a tight upper bound on the sum of sizes of cross $t$-intersecting separated families. Our result not only extends a previous theorem of Frankl, Liu, Wang and Yang on separated families, but also provides explicit counterexamples to an open problem proposed by them, thereby settling their problem in the negative.

math.CO↗
arXiv2025

Non-uniform pairwise cross $t$-intersecting families

Let $ n\geq t\geq 1$ and $ \mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}_m \subseteq 2^{[n]}$ be non-empty families. We say that they are pairwise cross $t$-intersecting if $|A_i\cap A_j|\geq t$ holds for any $A_i\in \mathcal{A}_i$ and $A_j\in \mathcal{A}_j$ with $i\neq j$. In the case where $m=2$ and $\mathcal{A}_1=\mathcal{A}_2$, determining the maximum size $M(n,t)$ of a non-uniform $t$-intersecting family of sets over $[n]$ was solved by Katona (1964), and enhanced by Frankl (2017), and recently by Li and Wu (2024). In this paper, we establish the following upper bound: if $ \mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}_m \subseteq 2^{[n]}$ are non-empty pairwise cross $t$-intersecting families, then $$ \sum_{i=1}^m |\mathcal{A}_i| \leq \max \left\{ \sum_{k=t} ^{n}\binom{n}{k} + m - 1, \, m M(n, t) \right\}. $$ Furthermore, we provide a complete characterization of the extremal families that achieve the bound. Our result not only generalizes an old result of Katona (1964) for a single family, but also extends a theorem of Frankl and Wong (2021) for two families. Moreover, our result could be viewed as a non-uniform version of a recent theorem of Li and Zhang (2025). The key in our proof is to utilize the generating set method and the pushing-pulling method together.

math.CO↗
arXiv2025

Spectral extremal problems for non-bipartite graphs without odd cycles

A well-known result of Mantel asserts that every $n$-vertex triangle-free graph $G$ has at most $\lfloor n^2/4 \rfloor$ edges. Moreover, Erd\H{o}s proved that if $G$ is further non-bipartite, then $e(G)\le \lfloor {(n-1)^2}/{4}\rfloor +1$. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] established a spectral version by showing that if $G$ is a triangle-free non-bipartite graph on $n$ vertices, then $\lambda (G)\le \lambda (S_1(T_{n-1,2}))$, with equality if and only if $G=S_1(T_{n-1,2})$, where $S_1(T_{n-1,2})$ is obtained from $T_{n-1,2}$ by subdividing an edge. In this paper, we investigate the maximum spectral radius of a non-bipartite graph without some short odd cycles. Let $C_{2\ell +1}(T_{n-2\ell, 2})$ be the graph obtained by identifying a vertex of $C_{2\ell+1}$ and a vertex of the smaller partite set of $T_{n-2\ell ,2}$. We prove that for $1\le \ell < k$ and $n\ge 187k$, if $G$ is an $n$-vertex $\{C_3,\ldots ,C_{2\ell -1},C_{2k+1}\}$-free non-bipartite graph, then $\lambda (G)\le \lambda (C_{2\ell +1}(T_{n-2\ell, 2}))$, with equality if and only if $G=C_{2\ell +1}(T_{n-2\ell, 2})$. This result could be viewed as a spectral analogue of a min-degree result due to Yuan and Peng [European J. Combin. 127 (2025)]. Moreover, our result extends a result of Guo, Lin and Zhao [Linear Algebra Appl. 627 (2021)] as well as a recent result of Zhang and Zhao [Discrete Math. 346 (2023)] since we can get rid of the condition that $n$ is sufficiently large. The argument in our proof is quite different and makes use of the classical spectral stability method and the double-eigenvector technique. The main innovation lies in a more clever argument that guarantees a subgraph to be bipartite after removing few vertices, which may be of independent interest.

math.CO↗
arXiv2024

Proof of Frankl's conjecture on cross-intersecting families

Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if for every $F\in \mathcal{F}$ and $G\in \mathcal{G}$, the intersection $F\cap G$ is non-empty. For any positive integers $n$ and $k$, let $\binom{[n]}{k}$ denote the family of all $k$-element subsets of $\{1,2,\ldots,n\}$. Let $t, s, k, n$ be non-negative integers with $k \geq s+1$ and $n \geq 2 k+t$. In 2016, Frankl proved that if $\mathcal{F} \subseteq\binom{[n]}{k+t}$ and $\mathcal{G} \subseteq\binom{[n]}{k}$ are cross-intersecting families, and $\mathcal{F}$ is $(t+1)$-intersecting and $|\mathcal{F}| \geq 1$, then $|\mathcal{F}|+|\mathcal{G}| \leq\binom{n}{k}-\binom{n-k-t}{k}+1$. Furthermore, Frankl conjectured that under an additional condition $\binom{[k+t+s]} {k+t}\subseteq\mathcal{F}$, the following inequality holds: $$ |\mathcal{F}|+|\mathcal{G}| \leq\binom{k+t+s}{k+t}+\binom{n}{k}-\sum_{i=0}^s\binom{k+t+s}{i}\binom{n-k-t-s}{k-i}. $$ In this paper, we prove this conjecture. The key ingredient is to establish a theorem for cross-intersecting families with a restricted universe. Moreover, we derive an analogous result for this conjecture.

math.CO↗