Zeyu Zheng

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 is a $\ell$ is a prime, the maximum size of an $\ell$-Oddtown is $n$. For 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 improve this to $ωn-(2ω+\varepsilon)\log_2 n$ for most $\ell$ and $n$ using a combination of linear algebraic and Fourier-analytic arguments.

We study the problem of finding the largest number $T(n, m)$ of ternary vectors of length $n$ such that for any three distinct vectors there are at least $m$ coordinates where they pairwise differ. For $m = 1$, this is the classical trifference problem which is wide open. We prove upper and lower bounds on $T(n, m)$ for various ranges of the parameter $m$ and determine the phase transition threshold on $m=m(n)$ where $T(n, m)$ jumps from constant to exponential in $n$. By relating the linear version of this problem to a problem on blocking sets in finite geometry, we give explicit constructions and probabilistic lower bounds. We also compute the exact values of this function and its linear variation for small parameters.

We investigate the extremal properties of saturated partial plane embeddings of maximal planar graphs. For a planar graph $G$, the plane-saturation number $\mathrm{sat}_{\mathcal{P}}(G)$ denotes the minimum number of edges in a plane subgraph of $G$ such that the addition of any edge either violates planarity or results in a graph that is not a subgraph of $G$. We focus on maximal planar graphs and establish an upper bound on $\mathrm{sat}_{\mathcal{P}}(G)$ by showing there exists a universal constant $ε> 0$ such that $\mathrm{sat}_{\mathcal{P}}(G) < (3-ε)v(G)$ for any maximal planar graph $G$ with $v(G) \geq 16$. This answers a question posed by Clifton and Simon. Additionally, we derive lower bound results and demonstrate that for maximal planar graphs with sufficiently large number of vertices, the minimum ratio $\mathrm{sat}_{\mathcal{P}}(G)/e(G)$ lies within the interval $(1/16, 1/9 + o(1)]$.

The $\textit{Abelian Sandpile}$ model is a well-known model used in exploring $\textit{self-organized criticality}$. Despite a large amount of work on other aspects of sandpiles, there have been limited results in efficiently computing the terminal state, known as the $\textit{sandpile prediction}$ problem. On graphs with special structures, we present algorithms that compute the terminal configurations for sandpile instances in $O(n \log n)$ time on trees and $O(n)$ time on paths, where $n$ is the number of vertices. Our algorithms improve the previous best runtime of $O(n \log^5 n)$ on trees [Ramachandran-Schild SODA '17] and $O(n \log n)$ on paths [Moore-Nilsson '99]. To do so, we move beyond the simulation of individual events by directly computing the number of firings for each vertex. The computation is accelerated using splittable binary search trees. In addition, we give algorithms in $O(n)$ time on cliques and $O(n \log^2 n)$ time on pseudotrees. On general graphs, we propose a fast algorithm under the setting where the number of chips $N$ could be arbitrarily large. We obtain a $\log N$ dependency, improving over the $\mathtt{poly}(N)$ dependency in purely simulation-based algorithms. Our algorithm also achieves faster performance on various types of graphs, including regular graphs, expander graphs, and hypercubes. We also provide a reduction that enables us to decompose the input sandpile into several smaller instances and solve them separately.