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Ting-Wei Chao

Publications and source records attributed to Ting-Wei Chao.

At least 19 recordsLinked to original sources
arXiv2026

The Oddtown problem modulo a composite number

A family of subsets $\mathcal{A}$ of an $n$-element set is called an $\ell$-Oddtown if the sizes of all sets are not divisible by $\ell$, but the sizes of pairwise intersections are divisible by $\ell$. Berlekamp and Graver showed that when $\ell$ is a prime, the maximum size of an $\ell$-Oddtown is $n$. Babai and Frankl extended this to prime powers, and asked whether the maximum size is still $n$ when $\ell$ is not a prime power, a question that was open even for $\ell=6$. For square-free composite moduli with $ω$ distinct prime factors, the argument of Szegedy gives an upper bound of $ωn-ω\log_2 n$ on the size of an $\ell$-Oddtown. We answer the question of Babai and Frankl in the negative by constructing $\ell$-Oddtowns of size $ωn-o(n)$, which shows that the leading term $ωn$ cannot be improved. We also improve Szegedy's upper bound to $ωn-(2ω+\varepsilon)\log_2 n$ for most $\ell$ and $n$ using a combination of linear algebraic and Fourier-analytic arguments.

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arXiv2026

Asymptotically sharp bounds for affine subspace statistics in $\mathbb F_2^n$

Given a subset $A \subseteq \mathbb F_2^n$, we can consider the distribution of the intersection size of $A$ with a uniformly random $d$-flat $F$. Motivated by the edge statistics problem and the hypercube statistics problem, the affine subspace statistics problem concerns the maximum of $\mathbb{P}[|F\cap A|=s]$ among $A \subseteq \mathbb F_2^n$ for any fixed $s\in\{1,\dots,2^d\}$ over a uniformly random $d$-flat $F$. We use $λ^*(d,s)$ to denote the limit of the maximum when $n$ goes to infinity. In this note, we prove tight bounds for $λ^*(d,s)$ in two different regimes. For $s=j2^k$ where $j$ is a positive odd integer, the best known lower bound construction achieving $λ^*(d,s)\ge 1-2^{-k}$ is due to taking $A$ as the union of $j$ parallel $(n-d+k)$-flats in $\mathbb F_2^n$. Our main result is a matching upper bound with an additive error term of $O(2^{-3k/2})$. We also study the case $s=1$, where we determine $λ^*(d,1)$ exactly. We show that the random construction where each point is included with probability $2^{-d}$ is optimal.

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arXiv2026

Vertex-minor universality of a random graph

Given a graph $G$ and a vertex $v\in V(G)$, a local complementation at $v$ on $G$ is an operation that replaces the induced graph on the neighborhood of $v$ by its complement. A graph $H$ is a vertex-minor if $H$ can be obtained from $G$ by a sequence of vertex deletions and local complementation. A graph is said to be $k$-vertex-minor universal if it contains every $k$-vertex graph on any $k$-subset of vertices as a vertex minor. Previously, Ascoli--Fredrickson--Fredrickson--McFarland--Post proved that with high probability $G(n,1/2)$ is $Ω(\sqrt{n})$-vertex-minor universal. Furthermore, they conjectured that with high probability $G(n,p)$ and $G(n,1-p)$ are $Ω(p\sqrt{n})$-vertex-minor universal for all $ω(1/\sqrt{n})\le p\le 1/2$. In this short note, we confirm this conjecture up to an extra logarithm factor and show that this is true with probability $1-2^{-Ω(p^2n)}$ if $Ω(\log n/\sqrt{n})\le p\le 1/2$. Together with a complementary result which applies to the regime where $1/\sqrt{n}\le p\le n^{-1/3}$ produced by an internal model at OpenAI, the conjecture is fully confirmed.

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arXiv2026

Even-degeneracy of a random graph

A graph is even-degenerate if one can iteratively remove a vertex of even degree at each step until at most one edge remains. Recently, Janzer and Yip showed that the Erdős--Renyi random graph $G(n,1/2)$ is even-degenerate with high probability, and asked whether an analogous result holds for any general $G(n,p)$. In this paper, we answer this question for any constant $p\in (0,1)$ in affirmation by proving that $G(n,p)$ is even-degenerate with high probability.

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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.

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arXiv2026

Uniform Set Systems with Uniform Witnesses

Frankl--Pach and Erdős conjectured that any $(d+1)$-uniform set family $\mathcal{F}\subseteq \binom{[n]}{d+1}$ with VC-dimension at most $d$ has size at most $\binom{n-1}{d}$ when $n$ is sufficiently large. Ahlswede and Khachatrian showed that the conjecture is false by giving a counterexample of size $\binom{n-1}{d}+\binom{n-4}{d-2}$. For a set family $\mathcal{F}\subseteq \binom{[n]}{d+1}$, the condition that its VC-dimension is at most $d$ can be reformulated as follows: for any $F\in\mathcal{F}$, there exists a set $B_F\subseteq F$ such that $F\cap F'\neq B_F$ for all $F'\in\mathcal{F}$. In this direction, the first author, Xu, Yip, and Zhang conjectured that the bound $\binom{n-1}{d}$ holds if we further assume that $|B_F|=s$ for every $F\in \mathcal{F}$ and for some fixed $0\leq s\leq d$. The case $s=0$ is exactly the Erdős--Ko--Rado theorem, and the cases $s\in \{1,d\}$ were proved in the paper by the first author, Xu, Yip, and Zhang. In this short note, we show that the conjecture holds when $s\leq d/2$, and the maximal constructions are stars. Moreover, we construct non-star set families of size $\binom{n-1}{d}$ satisfying the condition for $d/2<s\leq d-1$, which suggests that the problem is substantially different in these cases.

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arXiv2026

Finite field Nikodym problem for spread line sets

A set of points $N\subseteq \mathbb{F}_q^d$ is a Nikodym set if, for any $x\in \mathbb{F}_q^d$, there is a line $\ell$ through $x$ such that $\ell\setminus\{x\}\subseteq N$. We conjecture that $|N|=q^d-O_d(q^{d/(d-1)})$ and prove it under an extra algebraic assumption.

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arXiv2026

When entropy meets Turán: new proofs and hypergraph Turán results

In this paper, we provide a new proof of a density version of Turán's theorem. We also rephrase both the theorem and the proof using entropy. With the entropic formulation, we show that some naturally defined entropic quantity is closely connected to other common quantities such as Lagrangian and spectral radius. In addition, we also determine the Turán density for a new family of hypergraphs, which we call tents. Our result can be seen as a new generalization of Mubayi's result on the extended cliques.

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arXiv2025

Edge inducibility via local directed graphs

In this paper we introduce the edge inducibility problem. This is a common refinement of both the well known Kruskal--Katona theorem and the inducibility question introduced by Pippenger and Golumbic. Our first result is a hardness result. It shows that for any graph $G$, there is a related graph $G'$ whose edge inducibility determines the vertex inducibility of $G$. Moreover, we determine the edge inducibility of every $G$ with at most $4$ vertices, and make some progress on the cases $G=C_5,P_6$. Lastly, we extend our hardness result to graphs with a perfect matching that is the unique fractional perfect matching. This is done by introducing locally directed graphs, which are natural generalizations of directed graphs.

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arXiv2025

Uniform set systems with small VC-dimension

We investigate the longstanding problem of determining the maximum size of a $(d+1)$-uniform set system with VC-dimension at most $d$. Since the seminal 1984 work of Frankl and Pach, which established the elegant upper bound $\binom{n}{d}$, this question has resisted significant progress. The best-known lower bound is $\binom{n-1}{d} + \binom{n-4}{d-2}$, obtained by Ahlswede and Khachatrian, leaving a substantial gap of $\binom{n-1}{d-1}-\binom{n-4}{d-2}$. Despite decades of effort, improvements to the Frankl--Pach bound have been incremental at best: Mubayi and Zhao introduced an $Ω_d(\log{n})$ improvement for prime powers $d$, while Ge, Xu, Yip, Zhang, and Zhao achieved a gain of 1 for general $d$. In this work, we provide a purely combinatorial approach that significantly sharpens the Frankl--Pach upper bound. Specifically, for large $n$, we demonstrate that the Frankl--Pach bound can be improved to $\binom{n}{d} - \binom{n-1}{d-1} + O_d(n^{d-1 - \frac{1}{4d-2}})=\binom{n-1}{d}+O_d(n^{d-1 - \frac{1}{4d-2}})$. This result completely removes the main term $\binom{n-1}{d-1}$ from the previous gap between the known lower and upper bounds. It also offers fresh insights into the combinatorial structure of uniform set systems with small VC-dimension. In addition, the original Erdős--Frankl--Pach conjecture, which sought to generalize the EKR theorem in the 1980s, has been disproven. We propose a new refined conjecture that might establish a sturdier bridge between VC-dimension and the EKR theorem, and we verify several specific cases of this conjecture, which is of independent interest.

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arXiv2024

Many cliques with small degree powers

Suppose $0 < p \le \infty$. For a simple graph $G$ with a vertex-degree sequence $d_1, \dots, d_n$ satisfying $(d_1^p + \dots + d_n^p)^{1/p} \le C$, we prove asymptotically sharp upper bounds on the number of $t$-cliques in $G$. This result bridges the $p = 1$ case, which is the notable Kruskal--Katona theorem, and the $p = \infty$ case, known as the Gan--Loh--Sudakov conjecture, and resolved by Chase. In particular, we demonstrate that the extremal construction exhibits a dichotomy between a single clique and multiple cliques at $p_0 = t - 1$. Our proof employs the entropy method.

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arXiv2024

When Joints Meet Extremal Graph Theory: Hypergraph Joints

The Kruskal--Katona theorem determines the maximum number of $d$-cliques in an $n$-edge $(d-1)$-uniform hypergraph. A generalization of the theorem was proposed by Bollobás and Eccles, called the partial shadow problem. The problem asks to determine the maximum number of $r$-sets of vertices that contain at least $d$ edges in an $n$-edge $(d-1)$-uniform hypergraph. In our previous work, we obtained an asymptotically tight upper bound via its connection to the joints problem, a problem in incidence geometry. In a different direction, Friedgut and Kahn generalized the Kruskal--Katona theorem by determining the maximum number of copies of any fixed hypergraph in an $n$-edge hypergraph, up to a multiplicative factor. In this paper, using the connection to the joints problem again, we generalize our previous work to show an analogous partial shadow phenomenon for any hypergraph, generalizing Friedgut and Kahn's result. The key idea is to encode the graph-theoretic problem with new kinds of joints that we call hypergraph joints. Our main theorem is a generalization of the joints theorem that upper bounds the number of hypergraph joints, which the partial shadow phenomenon immediately follows from. In addition, with an appropriate notion of multiplicities, our theorem also generalizes a generalization of Hölder's inequality considered by Finner.

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arXiv2024

Empty red-red-blue triangles

Let $P$ be a $2n$-point set in the plane that is in general position. We prove that every red-blue bipartition of $P$ into $R$ and $B$ with $|R| = |B| = n$ generates $Ω(n^{3/2})$ red-red-blue empty triangles.

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arXiv2024

Kruskal--Katona-Type Problems via the Entropy Method

In this paper, we investigate several extremal combinatorics problems that ask for the maximum number of copies of a fixed subgraph given the number of edges. We call problems of this type Kruskal--Katona-type problems. Most of the problems that will be discussed in this paper are related to the joints problem. There are two main results in this paper. First, we prove that, in a $3$-edge-colored graph with $R$ red, $G$ green, $B$ blue edges, the number of rainbow triangles is at most $\sqrt{2RGB}$, which is sharp. Second, we give a generalization of the Kruskal--Katona theorem that implies many other previous generalizations. Both arguments use the entropy method, and the main innovation lies in a more clever argument that improves bounds given by Shearer's inequality.

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arXiv2023

Tight Bound and Structural Theorem for Joints

A joint of a set of lines $\mathcal{L}$ in $\mathbb{F}^d$ is a point that is contained in $d$ lines with linearly independent directions. The joints problem asks for the maximum number of joints that are formed by $L$ lines. Guth and Katz showed that the number of joints is at most $O(L^{3/2})$ in $\mathbb{R}^3$ using polynomial method. This upper bound is met by the construction given by taking the joints and the lines to be all the $d$-wise intersections and all the $(d-1)$-wise intersections of $M$ hyperplanes in general position. Furthermore, this construction is conjectured to be optimal. In this paper, we verify the conjecture and show that this is the only optimal construction by using a more sophisticated polynomial method argument. This is the first tight bound and structural theorem obtained using this method. We also give a new definition of multiplicity that strengthens the main result of a previous work by Tidor, Zhao and the second author. Lastly, we relate the joints problem to some set-theoretic problems and prove conjectures of Bollobás and Eccles regarding partial shadows.

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arXiv2022

Digital almost nets

Digital nets (in base $2$) are the subsets of $[0,1]^d$ that contain the expected number of points in every not-too-small dyadic box. We construct sets that contain almost the expected number of points in every such box, but which are exponentially smaller than the digital nets. We also establish a lower bound on the size of such almost nets.

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arXiv2022

A simple proof of the Gan-Loh-Sudakov conjecture

We give a new unified proof that any simple graph on $n$ vertices with maximum degree at most $Δ$ has no more than $a\binom{Δ+1}{t}+\binom{b}{t}$ cliques of size $t \ (t \ge 3)$, where $n = a(Δ+1)+b \ (0 \le b \le Δ)$.

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