arXiv subjects

Xizhi Liu

Publications and source records attributed to Xizhi Liu.

At least 19 recordsLinked to original sources

arXiv2026

The Erdős Matching Conjecture for 4-uniform hypergraphs

We prove the Erdős Matching Conjecture for $4$-uniform hypergraphs for every matching number $s\ge6004$. Specifically, for integers $s\ge 6004$ and $n\ge 4s+4$, if a $n$-vertex 4-uniform hypergraph $\mathcal F$ does not contain a matching of size $s+1$, then \[ |\mathcal F|\le \max\left\{\binom{4s+3}{4},\binom n4-\binom{n-s}{4}\right\}. \] Our proof introduces a finite-board reduction for general uniformity. It reduces the global conjecture to a lower-uniformity bound and a fixed finite optimization at the two adjacent vertex numbers where the candidate constructions exchange dominance. In the $4$-uniform case the board has $19$ vertices, and its weighted inequality splits into $19$-, $15$-, and $11$-vertex layers. The hardest layer is resolved by exact rational dual certificates and deterministic integer searches. All computer-assisted steps are checked in exact arithmetic by verifiers that reconstruct the finite systems directly from their mathematical definitions.

math.CO↗

arXiv2026

Turing universality, computability, and incompleteness in hypergraph Turán theory

Given a finite family $\mathcal F$ of forbidden $r$-graphs, the Turán problem asks for the maximum asymptotic edge density of $\mathcal F$-free $r$-graphs and the structure of near-extremal examples. We show that both questions can encode arbitrary computation. Fix a universal Turing machine $\mathsf U$. For every sufficiently large fixed $r$, there is a rational $τ_r\in(0,1)$ such that, from each binary word $β$, one can construct a finite family $\mathcal F_{r,β}$ with $π(\mathcal F_{r,β})=τ_r$ if $\mathsf U$ does not halt on $β$, and $π(\mathcal F_{r,β})>τ_r$ otherwise. The same dichotomy governs extremal structure. We construct finite families $\mathcal G_{r,β}$ such that nonhalting gives a unique extremal limit and Erdős--Simonovits stability, whereas halting gives two nonempty compact extremal phases separated by the sign of a fixed continuous statistic. Hence uniqueness and connectedness of the extremal space, symmetry breaking, two-phase behavior, and stability are all undecidable. The reductions are effective and verifiable in ZFC by finite certificates. Consequently, for every consistent computably axiomatized extension of ZFC and every sufficiently large fixed $r$, there is a finite family $\mathcal F$ for which the true equality $π(\mathcal F)=τ_r$ is neither provable nor refutable; analogous independence holds for the five structural properties above. We also obtain effective approximation, classify exact comparison complexity, and show that the smallest improvement witnesses have Busy-Beaver growth, with no uniform computable positive lower bound on the density gain.

math.CO↗

arXiv2026

Covering large-dimensional Euclidean spaces by random translates of a given convex body

Determining the minimum density of a covering of $\mathbb{R}^{n}$ by Euclidean unit balls as $n\to\infty$ is a major open problem, with the best known results being the lower bound of $\left(\mathrm{e}^{-3/2}+o(1)\right)n$ by Coxeter, Few and Rogers [Mathematika 6, 1959] and the upper bound of $\left(1/2+o(1) \right)n \ln n$ by Dumer [Discrete Comput. Geom. 38, 2007]. We prove that there are ball coverings of $\mathbb{R}^n$ attaining the asymptotically best known density $\left(1/2+o(1) \right)n \ln n$ such that, additionally, every point of $\mathbb{R}^n$ is covered at most $\left(1.79556... + o(1)\right) n \ln n$ times. This strengthens the result of Erdős and Rogers [Acta Arith. 7, 1961/62] who had the maximum multiplicity at most $\left(\mathrm{e} + o(1)\right) n \ln n$. On the other hand, we show that the method that was used for the best known ball coverings (when one takes a random subset of centres in a fundamental domain of a suitable lattice in $\mathbb{R}^n$ and extends this periodically) fails to work if the density is less than $(1/2+o(1))n\ln n$; in fact, this result remains true if we replace the ball by any convex body $K$. Also, we observe that a ``worst'' convex body $K$ here is a cube, for which the packing density coming from random constructions is only $(1+o(1))n\ln n$.

math.CO↗

arXiv2026

Sabotaging Mantel's Theorem

One of the earliest results in extremal graph theory, Mantel's theorem, states that the maximum number of edges in a triangle-free graph $G$ on $n$ vertices is $\lfloor n^2/4 \rfloor$. We investigate how this extremal bound is affected when $G$ is additionally required to contain a prescribed graph $H$ as a subgraph. We establish general upper and lower bounds for this problem, which are tight in the exponent for random triangle-free graphs and graphs generated by the triangle-free process, when the size of $H$ lies within certain ranges.

math.CO↗

arXiv2026

Tight Hamiltonian Cycles in Uniformly Dense $3$-Graphs

We study minimum degree conditions for tight Hamiltonian cycles in uniformly dense $3$-uniform hypergraphs. We prove that for every $d,α>0$, every sufficiently large $(ρ,d)$-dense $3$-graph on $n$ vertices with minimum codegree at least $(1/3+α)n$ contains a tight Hamiltonian cycle. This resolves a problem of Aigner-Horev and Levy in a stronger form, and the constant $1/3$ is asymptotically best possible. We also show that uniform density does not lower the asymptotic vertex-degree threshold: there are $(ρ,d)$-dense $3$-graphs with minimum vertex degree $(5/9-o(1))\binom{n}{2}$ and no tight Hamiltonian cycle. Finally, we construct $(ρ,2-\sqrt{3})$-dense examples with minimum codegree $(2-\sqrt{3}-o(1))n$ and no tight Hamiltonian cycle, answering negatively a question of Ara{ú}jo, Piga and Schacht.

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arXiv2026

Nearly Sharp Bounds for Lattice Coverings by Convex Bodies

For an $n$-dimensional convex body $K$, let $θ_L(K)$ denote its lattice covering density, and let $Θ_L^{\mathrm{conv}}(n)$ and $Θ_L^{\mathrm{sym}}(n)$ be the corresponding worst-case quantities over all convex bodies and over origin-symmetric convex bodies, respectively. Before this work, these quantities were known only to lie between a lower bound of order $n$ and an upper bound of order $n^2$, so even their polynomial order was undetermined. We prove that there are absolute constants $c,C>0$ such that \[ c n\log n \le Θ_L^{\mathrm{sym}}(n) \le Θ_L^{\mathrm{conv}}(n) \le Cn\log n\,(\log\log n)^{10/3+o(1)}. \] Thus both worst-case quantities are $n\log n\,(\log n)^{o(1)}$, and the upper and lower bounds differ by a factor at most $(\log\log n)^{10/3+o(1)}$. For the upper bound, a vertical--horizontal amplification based on weighted Boolean cubes combines covering estimates for low-codimensional sections into an exact lattice covering of an arbitrary convex body. For the lower bound, a random-slab construction and Poisson witnesses on flat tori show, with positive probability, that the resulting body admits no lattice covering of density below $c n\log n$.

math.MG↗

arXiv2026

Forcing Quasirandomness via Rooted F-Densities

Let $F$ be a finite graph with at least one edge, and let $W$ be a graphon. We show that if the density of $F$ rooted at each edge is almost everywhere constant, then either $t(F,W)=0$ or $W$ is constant. For edge-transitive $F$, one rooted equation suffices. This recovers the edge-rooted triangle theorem of Reiher and Schacht. In their terminology, our result also shows that every clique is $2$-forcing, answering a question they posed. We give an explicit stability estimate when $W$ is bounded away from zero. Our proof has two steps: an entropy argument turns constant rooted densities into an additive identity for $\log W$, and a Hoeffding decomposition determines all solutions of that identity. The same method gives exact classifications and quantitative stability estimates for symmetric uniform hyperkernels, dissociated Aldous--Hoover hypergraphons, directed kernels, and tournamentons.

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arXiv2026

Intervals of uniform Turán densities

We prove that the set $Π_{\therefore,\infty}$ of uniform Turán densities of possibly infinite families of $3$-graphs contains a terminal interval: there exists $δ>0$ such that $[1-δ,1]\subseteqΠ_{\therefore,\infty}$. Consequently, $Π_{\therefore,\infty}$ has positive Lebesgue measure and Hausdorff dimension $1$.

math.CO↗

arXiv2026

The inducibility of 6-vertex graphs

The inducibility constant $λ_{F}$ of a graph $F$ is the asymptotically maximum induced density of $F$ in a growing sequence of graphs. This paper systematically investigates the case when $F$ has 6 vertices (and there are 78 cases to consider up to isomorphism and complementation). We show that flag algebras can compute the sharp upper bound on $λ_F$ in 36 cases of which, as far as the authors know, 30 are new results. In each of the solved cases, we also prove results about the structure of large (almost) extremal graphs. In particular, we establish perfect stability in all 32 cases when the extremal construction has no quasirandom parts. We also present conjectures about the value of $λ_{F}$ for 12 further cases (where the upper and lower bounds are very close to each other).

math.CO↗

arXiv2026

Strong counterexamples to Mubayi's supersaturation conjecture in every uniformity

The supersaturation problem asks, for a fixed $r$-graph $\mathcal F$, for the minimum number of copies of $\mathcal F$ in an $n$-vertex $r$-graph with $\ex(n,\mathcal F)+q$ edges. Mubayi conjectured a local form of supersaturation under a stability hypothesis: if $\mathcal F$ is non-$r$-partite and stable, meaning roughly that the extremal $\mathcal F$-free construction is unique and all near-extremal $\mathcal F$-free $r$-graphs are close to it, then this minimum should be at least $q c(n,\mathcal F)$, where $c(n,\mathcal F)$ is the minimum number of copies created by adding one edge to the extremal $\mathcal F$-free $r$-graph. We disprove this conjectured local lower bound in every uniformity. For every $r\ge2$ and every $K>1$, we construct a stable $r$-graph $\mathcal F$ such that, for all sufficiently large $n$ and every $1\le q\le δn$, there is an $n$-vertex $r$-graph with $\ex(n,\mathcal F)+q$ edges and at most $K^{-1}q c(n,\mathcal F)$ copies of $\mathcal F$. Thus the conjectured lower bound can already fail at $q=1$, and the failure can be by an arbitrarily large constant factor in every uniformity.

math.CO↗

arXiv2026

On Mubayi's Polynomial-Ideal Conjecture and a Hypergraph Turán Theorem

Among the many proofs of Turán's classical theorem, one particularly surprising proof due to Li and Li uses ideals in polynomial rings to record missing edges. Motivated by their proof, Mubayi proposed a hypergraph analogue, conjecturing that an ideal generated by multipartite $3$-graphs coincides with a differentiated diagonal-vanishing ideal. If true, this conjecture would imply the extremal-number part of Mubayi's classical hypergraph extension of Turán's theorem. We disprove this conjecture throughout the nontrivial parameter range. We then give an alternative algebraic proof of Mubayi's extremal formula for the family $\mathcal{K}_{\ell}^{(r)}$ of clique expansions, using monomial cover ideals and a Hilbert-function symmetrization theorem for square-zero quadratic monomial quotients.

math.CO↗

arXiv2026

Sharp bounds for minimal dependencies of linear-form powers

Motivated by the dimension-bound part of a problem of Bukh, we study Veronese circuits: how large can the span of $t$ linear forms be if their $m$-th powers are minimally linearly dependent? We prove the sharp finite dimension bound \[ \dim L\leq \frac{t+m-2}{m}. \] Here $\ell_1,\ldots,\ell_t$ are nonzero homogeneous linear forms over a field of characteristic zero, the powers $\ell_1^m,\ldots,\ell_t^m$ form a circuit, and $L=\Span\{\ell_1,\ldots,\ell_t\}$. Rational-normal-curve configurations attain equality for infinitely many pairs $(t,\dim L)$; in particular, the affine bound itself is sharp and the optimal leading constant in Bukh's question is $1/m$. The proof uses a coding-theoretic translation: the coefficient row space of the powers is the $m$-th Schur power of the coefficient code, and the minimality hypothesis makes this Schur power a full-support hyperplane to which the Schur-product Kneser theorem of Mirandola and Zémor applies. The same method yields flat-concentration and interpolation criteria, a Cayley--Bacharach lower bound, Segre--Veronese and positive-characteristic variants, and Hilbert-function constraints for equality and near-equality in Veronese circuits.

math.CO↗

arXiv2026

Turán numbers of $4$-uniform tight even cycles minus one edge

For every integer $k \ge 1$ and sufficiently large $n$, we show that the extremal construction for the Turán number of the $4$-uniform tight cycle of length $4k+2$ minus one edge is a complete odd-bipartite $4$-graph. In particular, since $C_{6}^{4-}$ contains the $4$-uniform expanded triangle as a subgraph, our result extends that of Frankl and Keevash--Sudakov on the Turán density and the Turán number of the $4$-uniform expanded triangle. We also show that the Turán density of $C_{4k+2}^{4}$ is $1/2$ for all integers $k \ge 2$, and establish the corresponding stability result. This strengthens the result of Sankar on the Turán density of $C_{4k+2}^{4}$ which holds only for all sufficiently large $k$.

math.CO↗

arXiv2026

Exact extremal constructions for the inducibility of blowup graphs

For a finite graph $H$ and a positive integer $h$, the $h$-blowup $H^{(h)}$ of $H$ is the graph obtained by replacing each vertex of $H$ by a set of size $h$ and each edge by a complete bipartite graph between the corresponding sets. We prove that, for every $H$, there exists a constant $h_*(H)$ such that whenever $h\ge h_*(H)$ and $n$ is sufficiently large, every $n$-vertex graph maximizing the number of induced copies of $H^{(h)}$ is a blowup of $H$. This refines the asymptotic result of Hatami, Hirst and Norine and settles the question posed by Bollobás, Egawa, Harris and Jin in 1995.

math.CO↗

arXiv2026

Upper Bounds on Turán Densities via Extremal Set Theory

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.

math.CO↗

arXiv2026

Vertex-colored Turán theorems with applications in extremal hypergraph problems

Balogh, Clemen, and Lidický proved that the $\ell_{2}$-norm Turán problem for $K_{5}^{3}$ is asymptotically solved by the balanced bipartite construction, and they further conjectured that this construction is uniquely extremal for all sufficiently large $n$. We confirm this conjecture. We also determine exactly the maximum number of cliques in an $n$-vertex $K_{5}^{3}$-free $3$-uniform hypergraph for all sufficiently large $n$, thereby verifying the corresponding case of a conjecture of Frankl, Gryaznov, and Talebanfard. The main ingredients are Turán-type theorems for vertex-colored graphs forbidding balanced cliques, including an edge bound, an $\ell_{2}$-norm bound, and a sharp crossing-triangle theorem in the two-colored balanced $K_{4}$-free case. We also use a local modification procedure within the stability method. This reduces the exact hypergraph problems to proving that the relevant objective function increases under suitable local changes near the bipartite construction.

math.CO↗

arXiv2026

Physical-Space Scarring in Generic Bunimovich Stadia

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.

math.AP↗

arXiv2026

The spectral inducibility of graphs

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 $r\le 2^a-1$.

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