arXiv subjects

Shuaichao Wang

Publications and source records attributed to Shuaichao Wang.

7 recordsLinked to original sources

arXiv2026

Independent Sets and Balanced Cycle-Linkings in 2-Connected Graphs

For every $α\ge3$ and all sufficiently large $n$, we identify a single graph that simultaneously maximizes the number of independent sets of every size among all $n$-vertex $2$-connected graphs with independence number $α$. This graph is unique up to isomorphism and is a balanced cycle-linking of the disjoint union of $α$ cliques whose orders differ by at most one. More precisely, for each $3\leβ\leα$, the graphs maximizing the number of independent $β$-sets are exactly the cycle-linkings whose clique-size cyclic words are $\lfloorβ/2\rfloor$-balanced. These results extend the corresponding extremal result for connected graphs to the $2$-connected setting. The proof combines generalized Turán-type clique counting and the edge-extremal theory of $2$-connected graphs with a coefficientwise balancing-switch argument. The switch also shows that a shortest imbalance of length $r$ first affects the independent-set count in degree $2r$.

math.CO↗

arXiv2026

A dichotomy for hypergraph Zarankiewicz problems on axis-parallel boxes

We study the Zarankiewicz problem for $r$-partite, $r$-uniform intersection hypergraphs arising from $r$ families of axis-parallel boxes in $\mathbb{R}^d$ with prescribed directions $F_1, \dots, F_r \subseteq \{1, \dots, d\}$. This extends the problems studied by Chan and Har-Peled on points and $d$-dimensional boxes in $\mathbb{R}^d$, corresponding to $(F_1,F_2)=(\varnothing,[d])$, as well as by Chan, Keller, and Smorodinsky on $r$ families of $d$-dimensional boxes, corresponding to $(F_1,\dots,F_r)=([d],\dots,[d])$. Our main result establishes a sharp dichotomy for the Zarankiewicz number in this setting: it is either $Θ_r(tn^{r-1})$ or at least $Ω\bigl( tn^{r-1} \cdot \frac{\log n}{\log\log n} \bigr)$, depending only on a simple set-theoretic condition on $(F_1,\dots,F_r)$, which we call $2$-coherence. Informally, $2$-coherence captures whether the configuration contains an underlying two-dimensional incidence structure, which is precisely what gives rise to the extra polylogarithmic factor. Our proof proceeds via a sequence of reductions and a geometric slicing argument that reduces the problem to planar incidence bounds.

math.CO↗

arXiv2025

Separating hypergraph Turán densities

Determining the Turán densities of hypergraphs is a notoriously difficult problem at the core of combinatorics. Although Turán posed this problem in 1941, $π(K_{\ell}^{(k)})$ remains unknown for all $\ell>k\geq 3$. Prior to this work, it was not even known whether $π(K_{\ell}^{(k)})<π(K_{\ell+1}^{(k)})$ holds for general $\ell$ and $k$, and the best-known bounds on $π(K_{\ell}^{(k)})$ are far from implying anything close to this. We prove that $π(K_{\ell}^{(k)})<π(K_{\ell+1}^{(k)})$, for all $\ell>k\geq 3$, and provide a general criterion to distinguish the Turán densities of two hypergraphs. As a corollary, we obtain that $π(K_{k+1}^{(k)})<π(K_{k+2}^{(k)-})$, for all $k\geq 3$. For $k=3$, this was previously proved by Markström, answering a question by Erdős.

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arXiv2024

On $3$-graphs with vanishing codegree Turán density

For a $k$-uniform hypergraph (or simply $k$-graph) $F$, the codegree Turán density $π_{\mathrm{co}}(F)$ is the supremum over all $α$ such that there exist arbitrarily large $n$-vertex $F$-free $k$-graphs $H$ in which every $(k-1)$-subset of $V(H)$ is contained in at least $αn$ edges. Recently, it was proved that for every $3$-graph $F$, $π_{\mathrm{co}}(F)=0$ implies $π_{\therefore}(F)=0$, where $π_{\therefore}(F)$ is the uniform Turán density of $F$ and is defined as the supremum over all $d$ such that there are infinitely many $F$-free $k$-graphs $H$ satisfying that any induced linear-size subhypergraph of $H$ has edge density at least $d$. In this paper, we introduce a layered structure for $3$-graphs which allows us to obtain the reverse implication: every layered $3$-graph $F$ with $π_{\therefore}(F)=0$ satisfies $π_{\mathrm{co}}(F)=0$. Along the way, we answer in the negative a question of Falgas-Ravry, Pikhurko, Vaughan and Volec [J. London Math. Soc., 2023] about whether $π_{\therefore}(F)\leqπ_{\mathrm{co}}(F)$ always holds. In particular, we construct counterexamples $F$ with positive but arbitrarily small $π_{\mathrm{co}}(F)$ while having $π_{\therefore}(F)\ge 4/27$.

math.CO↗

arXiv2023

Vanishing codegree Turán density implies vanishing uniform Turán density

For a $k$-uniform hypergraph (or simply $k$-graph) $F$, the codegree Turán density $π_{\mathrm{co}}(F)$ is the infimum over all $α$ such that any $n$-vertex $k$-graph $H$ with every $(k-1)$-subset of $V(H)$ contained in at least $αn$ edges has a copy of $F$. The uniform Turán density $π_{\therefore}(F)$ is the supremum over all $d$ such that there are infinitely many $F$-free $k$-graphs $H$ satisfying that any linear-size subhypergraph of $H$ has edge density at least $d$. Falgas-Ravry, Pikhurko, Vaughan and Volec [J. London Math. Soc., 2023] asked whether for every $3$-graph $F$, $π_{\therefore}(F)\leqπ_{\mathrm{co}}(F)$. We provide a positive answer to this question provided that $π_{\mathrm{co}}(F)=0$. Our proof relies on a random geometric construction and a new formulation of the characterization of $3$-graphs with vanishing uniform Turán density due to Reiher, R{ö}dl and Schacht [J. London Math. Soc., 2018]. Along the way, we answer a question of Falgas-Ravry, Pikhurko, Vaughan and Volec about subhypergraphs with linear minimum codegree in uniformly dense hypergraphs in the negative.

math.CO↗

arXiv2023

The degree and codegree threshold for generalized triangle and some trees covering

Given two $k$-uniform hypergraphs $F$ and $G$, we say that $G$ has an $F$-covering if for every vertex in $G$ there is a copy of $F$ covering it. For $1\leq i\leq k-1$, the minimum $i$-degree $δ_i(G)$ of $G$ is the minimum integer such that every $i$ vertices are contained in at least $δ_i(G)$ edges. Let $c_i(n,F)$ be the largest minimum $i$-degree among all $n$-vertex $k$-uniform hypergraphs that have no $F$-covering. In this paper, we consider the $F$-covering problem in $3$-uniform hypergraphs when $F$ is the generalized triangle $T$, where $T$ is a $3$-uniform hypergraph with the vertex set $\{v_1,v_2,v_3,v_4,v_5\}$ and the edge set $\{\{v_{1}v_{2}v_{3}\},\{v_{1}v_{2}v_{4}\},\{v_{3}v_{4}v_{5}\}\}$. We give the exact value of $c_2(n,T)$ and asymptotically determine $c_1(n,T)$. We also consider the $F$-covering problem in $3$-uniform hypergraphs when $F$ are some trees, such as the linear $k$-path $P_k$ and the star $S_k$. Especially, we provide bounds of $c_i(n,P_k)$ and $c_i(n,S_k)$ for $k\geq 3$, where $i=1,2$.

math.CO↗

arXiv2023

On extremal problems on multigraphs

An $(n,s,q)$-graph is an $n$-vertex multigraph in which every $s$-set of vertices spans at most $q$ edges. Erdős initiated the study of maximum number of edges of $(n,s,q)$-graphs, and the extremal problem on multigraphs has been considered since the 1990s. The problem of determining the maximum product of the edge multiplicities in $(n,s,q)$-graphs was posed by Mubayi and Terry in 2019. Recently, Day, Falgas-Ravry and Treglown settled a conjecture of Mubayi and Terry on the case $(s,q)=(4, 6a + 3)$ of the problem (for $a \ge 2$), and they gave a general lower bound construction for the extremal problem for many pairs $(s, q)$, which they conjectured is asymptotically best possible. Their conjecture was confirmed exactly or asymptotically for some specific cases. In this paper, we consider the case that $(s,q)=(5,\binom{5}{2}a+4)$ and $d=2$ of their conjecture, partially solve an open problem raised by Day, Falgas-Ravry and Treglown. We also show that the conjecture fails for $n=6$, which indicates for the case that $(s,q)=(5,\binom{5}{2}a+4)$ and $d=2$, $n$ needs to be sufficiently large for the conjecture to hold.

math.CO↗