arXiv subjects
Publications and source records attributed to Xizhi Liu.
We exhibit, in a systematic way, connections between hypergraph Turán problems and extremal set theory. More specifically, we construct natural families of uniform hypergraphs for which the upper bounds on their Turán densities reduce to classical problems in extremal set theory, including the Erdős--Ko--Rado theorem, $L$-intersecting families, and the Erdős matching problem.
For the family of Dirichlet stadia $S_t$ whose rectangular part has height $π$ and half-length $πt/2$, $t \in [1,2]$, we show that for Lebesgue almost every $t$ there exist real eigenfunctions $u_j$ and a smooth mean-zero physical observable $Q$ for which $\langle Q u_j,u_j\rangle$ has a non-zero subsequential limit. Consequently, along the same subsequence, the eigenfunction mass fails to equidistribute on a fixed region whose relative boundary in the interior of the stadium is smooth. This proves a physical-space strengthening of Hassell's non-QUE theorem for generic stadia, and thus gives an affirmative answer to Tao's question in Hassell's generic setting. The proof uses the classification of generic stadia in Hassell's argument. In each of the resulting cases, we construct an appropriate physical observable $Q$ that converts Hassell's phase-space obstruction to QUE into physical-space non-equidistribution.
We introduce a spectral version of the classical inducibility problem. Given an $\ell$-vertex graph $F$ and an $n$-vertex graph $G$, let $H_F(G)$ be the $\ell$-uniform hypergraph whose edges are the $\ell$-sets inducing a copy of $F$ in $G$. We study the maximum possible $α$-spectral radius of $H_F(G)$ over all $n$-vertex graphs $G$. For fixed $G$, this spectral parameter tends to $\ell!$ times the number of induced copies of $F$ in $G$ as $α\to\infty$, and therefore refines the usual induced-copy count. Our main result is a spectral analogue of the Brown--Sidorenko reduction: for every complete multipartite graph $F$, every $n$, and every $α\ge1$, a spectral extremal graph can be chosen to be complete multipartite. We also show that the leading asymptotic constant is the ordinary inducibility $i(F)$, and obtain exact multipartite reductions for stars $K_{1,t}$ and balanced complete $r$-partite graphs $K_{a,\ldots,a}$ with $r\le 2^a-1$.
We prove that, for every integer $r\ge 3$, the set $Π^{(r)}_\infty$ of Turán densities of (possibly infinite) families of $r$-graphs contains non-degenerate intervals, including an interval of the form $[1-δ_r,1]$ for some $δ_{r}>0$. This answers a question of Frankl, Peng, Rödl and Talbot from 2007. This also shows that the Hausdorff dimension of $Π^{(r)}_\infty$ has the maximum possible value 1, thus resolving a question of Grosu from 2016, whereas previously it was not even known whether it is non-zero. We also derive that the set of uniform Turán densities of finite families of $3$-graphs is dense in a non-degenerate interval.
The stability number of a forbidden family measures how many different structures are needed to approximate all near-extremal constructions avoiding it. An infinite stability number means that no finite list of structures suffices. We construct a simple explicit $3$-graph whose stability number is infinite. This extends the infinite-stability phenomenon for finite forbidden families, established by Hou--Li--Liu--Mubayi--Zhang, to the single-forbidden setting, and further develops the single-$3$-graph direction of Balogh--Clemen--Luo, in which exponentially many exact extremal constructions coexist with stability.
For an $r$-uniform hypergraph $G$, let $λ^{(p)}(G)$ denote its $p$-spectral radius, defined as the maximum of the polyform of $G$ over the unit sphere in the $\ell_p$-norm. Let $Q_k^r(n)$ be the complete $k$-chromatic $r$-graph on $n$ vertices with color classes as equal as possible. Kang--Nikiforov--Yuan conjectured that, for every $p\ge1$ and $n>(r-1)k$, the $r$-graph $Q_k^r(n)$ is the unique maximizer of $λ^{(p)}$ among all $k$-chromatic $r$-graphs of order $n$. They also conjectured the corresponding explicit bound \[ λ^{(p)}(G) \le r!\left(\tbinom nr-k\tbinom{n/k}{r}\right)n^{-r/p}, \] with equality only in the divisible extremal case. The case $r=3$ was established in their work. This paper resolves the remaining cases $r\ge4$, and hence settles both conjectures for all $r\ge3$. As a consequence, the same threshold gives an anti-Wilf-type spectral certificate: any $r$-graph of order $n$ whose $p$-spectral radius exceeds the displayed bound has chromatic number at least $k+1$.
We prove that $4/9$ is a non-jump for $3$-uniform hypergraphs. Our construction perturbs the $ABB$ pattern by inserting, inside the $B$-part, the union of a high-cogirth pair of Steiner triple systems. This goes below the barrier for non-jumps obtainable by Shaw's finite-pattern formulation of the Frankl--Rödl method introduced in 1984. All results employing this approach use patterns where one of the parts has complete shadow. As the $ABB$ pattern is the smallest one with this property, the value $4/9$ is the natural barrier using this technique, and we conjecture that $4/9$ is the smallest non-jump for $3$-graphs. If our conjecture is true, this would answer (in a very strong form) an old question of Erd\Hos.
Synchronization in dynamical systems on directed weighted networks is often associated with stronger coupling and denser interactions. This paper shows that the opposite can also occur: weakening selected edges may increase the generalized algebraic connectivity, denoted by $κ$, and in some nonlinear systems this spectral improvement is accompanied by a transition from nonsynchronization to synchronization. To explain this effect, we develop a perturbation-based spectral sensitivity framework for directed weighted networks. We derive an explicit first-order formula for the response of $κ$ to edge-weight perturbations and show that it decomposes into a directed cut-energy term and a stationary redistribution term. This decomposition clarifies how asymmetric flow structure and invariant-mass redistribution jointly determine the synchronization role of each edge. Based on this theory, we design sensitivity-guided algorithms for edge weakening, edge deletion, negative-edge insertion, and edge strengthening. Experiments on synthetic and real networks show that these methods identify critical edges whose modification yields substantial gains in $κ$. Simulations of first- and second-order nonlinear consensus dynamics further show markedly faster convergence and, in some cases, a transition from incoherence to synchronization. The results provide a local spectral mechanism by which reducing or reallocating coupling can enhance synchronization-related performance.
Motivated by the Erdős--Sós bipartite link conjecture, Füredi (Oberwolfach, 2004) asked for the asymptotic maximum edge density $π_{\mathrm{link}}(t)$ of $3$-graphs in which the link graph of every vertex is $t$-partite. Goldwasser's recursive blow-up construction based on projective planes gives the lower bound $π_{\mathrm{link}}(t)\ge 1-t^{-1}-(2+o_t(1))t^{-2}$ whenever $t-1$ is a prime power. In this note, we prove the upper bound $π_{\mathrm{link}}(t)\le 1-t^{-1}-t^{-2}/12$ for every $t \ge 2$. Together with Goldwasser's construction, this determines, up to a constant factor, the correct order of the gap between $π_{\mathrm{link}}(t)$ and the trivial averaging upper bound $1-t^{-1}$ for all prime-power values of $t-1$. In fact, our argument applies in the more general setting of $3$-graphs with no generalized daisies, equivalently, $3$-graphs in which the link graph of every vertex is $K_{t+1}$-free. We also establish an analogous upper bound for the positive $(r-1)$-codegree Turán density of generalized daisies.
A classical theorem of Ahlswede and Katona determines the maximum density of the $2$-edge star in a graph with a given edge density. Motivated by its application in hypergraph Turán problems, we establish a refinement of their result under the additional assumption that the graph contains a large independent set in which every vertex has high degree.
Let $I(F,n)$ denote the maximum number of induced copies of a graph $F$ in an $n$-vertex graph. The inducibility of $F$, defined as $i(F)=\lim_{n\to \infty} I(F,n)/\binom{n}{v(F)}$, is a central problem in extremal graph theory. In this work, we investigate the inducibility of Turán graphs $F$. This topic has been extensively studied in the literature, including works of Pippenger--Golumbic, Brown--Sidorenko, Bollobás--Egawa--Harris--Jin, Mubayi, Reiher, and the first author, and Yuster. Broadly speaking, these results resolve or asymptotically resolve the problem when the part sizes of $F$ are either sufficiently large or sufficiently small (at most four). We complete this picture by proving that for every Turán graph $F$ and sufficiently large $n$, the value $I(F,n)$ is attained uniquely by the $m$-partite Turán graph on $n$ vertices, where $m$ is given explicitly in terms of the number of parts and vertices of $F$. This confirms a conjecture of Bollobás--Egawa--Harris--Jin from 1995, and we also establish the corresponding stability theorem. Moreover, we prove an asymptotic analogue for $I_{k+1}(F,n)$, the maximum number of induced copies of $F$ in an $n$-vertex $K_{k+1}$-free graph, thereby completely resolving a recent problem of Yuster. Finally, our results extend to a broader class of complete multipartite graphs in which the largest and smallest part sizes differ by at most on the order of the square root of the smallest part size.
The relation between densities of cycles and the spectrum of a graphon, which implies that the spectra of convergent graphons converge, fundamentally relies on the self-adjointness of the linear operator associated with a graphon. In this short paper, we consider the setting of digraphons, which are limits of directed graphs, and prove that the spectra of convergent digraphons converge. Using this result, we establish the relation between densities of directed cycles and the spectrum of a digraphon.
We establish the following strengthening of the celebrated Andr{á}sfai--Erdős--Sós theorem: If $G$ is an $n$-vertex $K_{r+1}$-free graph whose minimum degree $δ(G)$ and maximum degree $Δ(G)$ satisfy \begin{align*} δ(G) > \min \left\{ \frac{3r-4}{3r-2}n-\frac{Δ(G)}{3r-2},~n-\frac{Δ(G)+1}{r-1} \right\}, \end{align*} then $G$ is $r$-partite. This bound is tight for all feasible values of $Δ(G)$. We also obtain an analogous tight result for graphs with large odd girth. Our proof does not rely on the Andr{á}sfai--Erdős--Sós theorem itself, and therefore yields an alternative proof of this classical result.
The famous Tetrahedron Conjecture of Turán from the 1940s asserts that the number of edges in an $n$-vertex $3$-graph without the tetrahedron, the complete $3$-graph on four vertices, cannot exceed that of the balanced complete cyclic $3$-partite $3$-graph, whose edges are of types $V_1 V_2 V_3$, $V_1 V_1 V_2$, $V_2 V_2 V_3$, and $V_3 V_3 V_1$. A recent surprising result of Balogh-Clemen-Lidický [J. Lond. Math. Soc. (2) 106 (2022)] shows that this conjecture is asymptotically true in the $\ell_2$-norm, where the number of edges is replaced by the sum of squared codegrees. They further conjectured that, in this $\ell_2$-norm setting, the $3$-partite construction is uniquely extremal for large $n$. We confirm this conjecture. Two key ingredients in our proofs include establishing a Mantel theorem for vertex-colored graphs that forbid certain types of triangles, and introducing a novel procedure integrated into Simonovits' stability method, which essentially reduces the task to verifying that the $\ell_2$-norm of certain near-extremal constructions increases under suitable local modifications. The strategy in the latter may be of independent interest and potentially applicable to other extremal problems.
A classical result of Sidorenko (1989) shows that the Turán density of every $r$-uniform hypergraph with three edges is bounded from above by $1/2$. For even $r$, this bound is tight, as demonstrated by Mantel's theorem on triangles and Frankl's theorem on expanded triangles. In this note, we prove that for odd $r$, the bound $1/2$ is never attained, thereby answering a question of Keevash and revealing a fundamental difference between hypergraphs of odd and even uniformity. Moreover, our result implies that the expanded triangles form the unique class of three-edge hypergraphs whose Turán density attains $1/2$.
We show that there exists a lattice covering of $\mathbb{R}^n$ by Eucledian spheres of equal radius with density $O\big(n \ln^β n \big)$ as $n\to\infty$, where \begin{align*} β:= \frac{1}{2} \log_2 \left(\frac{8 π\mathrm{e}}{3\sqrt 3}\right)=1.85837...\,. \end{align*} This improves upon the previously best known upper bound by Rogers from 1959 of $O\big(n \ln^α n \big)$, where $α:= \frac{1}{2} \log_{2}(2π\mathrm{e})=2.0471...\,.$
Combining two well-studied variants of the classical Turán problem, the generalized Turán problem and the spectral Turán problem, we introduce the spectral generalized Turán problem and establish a general theorem that extends the result of Keevash--Lenz--Mubayi~\cite{KLM14} on the spectral Turán problem in this broader setting. As a quick application, we obtain the spectral Erdős Pentagon Theorem. We also introduce the notion of entropic density for generalized Turán problems, and show that it coincides with the generalized spectral radius, extending a recent result of Chao--Hans on entropic Turán density.
Let the tight $\ell$-cycle minus one edge $C_\ell^{3-}$ be the $3$-graph on $\{1,\dots,\ell\}$ consisting of $\ell-1$ consecutive triples in the cyclic order. We show that, for every $\ell\ge 5$ not divisible by $3$, the Turán density of $C_{\ell}^{3-}$ is $1/4$ and also prove some finer structure results. This proves a conjecture of Mubayi--Sudakov--Pikhurko from 2011 and extends the results of Balogh--Luo [Combinatorica 44 (2024) 949--976] who established analogous claims for all sufficiently large $\ell$. Results similar to ours were independently obtained by Lidický--Mattes--Pfender [arXiv:2409.14257].