Improvements on Hippchen's Conjecture

对超满猜想的新改进

A new improvement to the Overfull Conjecture

本文改进了超满猜想的条件，证明在Δ(G)-5δ(G)/3≥(2n-7)/3时，所有满足该条件的Δ-临界图都是超满图。

通过KKL定理改进 Erdős-Kleitman 猜想

Improvement on the Erdős-Kleitman conjecture via the KKL theorem

本文通过连接Kahn, Kalai和Linial关于布尔函数影响的基石结果，改进了Erdős-Kleitman猜想的下界，提出新的更紧的下限表达式。

迈向Vizing猜想的一个常数因子

A constant-factor step towards Vizing's conjecture

本文改进了Clark-Suen关于图支配数的乘积下界，将常数因子从1/2提升至约0.5643，首次实现绝对常数改进。

Small Cores in 3-uniform Hypergraphs

The main result of this paper is that for any $c>0$ and for large enough $n$ if the number of edges in a 3-uniform hypergraph is at least $cn^2$ then there is a core (subgraph with minimum degree at least 2) on at most 15 vertices. We conjecture that our result is not sharp and 15 can be replaced by 9. Such an improvement seems to be out of reach, since it would imply the following case of a long-standing conjecture by Brown, Erd\H os, and Sós; if there is no set of 9 vertices that span at least 6 edges of a 3-uniform hypergraph then it is sparse.

Improving the $\frac{1}{3}-\frac{2}{3}$ Conjecture for Width Two Posets

Extending results of Linial (1984) and Aigner (1985), we prove a uniform lower bound on the balance constant of a poset $P$ of width $2$. This constant is defined as $δ(P) = \max_{(x, y)\in P^2}\min\{\mathbb{P}(x\prec y), \mathbb{P}(y\prec x)\}$, where $\mathbb{P}(x\prec y)$ is the probability $x$ is less than $y$ in a uniformly random linear extension of $P$. In particular, we show that if $P$ is a width $2$ poset that cannot be formed from the singleton poset and the three element poset with one relation using the operation of direct sum, then \[δ(P)\ge\frac{-3 + 5\sqrt{17}}{52}\approx 0.338…

An improvement on the maximum number of $k$-Dominating Independent Sets

Erdős and Moser raised the question of determining the maximum number of maximal cliques or equivalently, the maximum number of maximal independent sets in a graph on $n$ vertices. Since then there has been a lot of research along these lines. A $k$-dominating independent set is an independent set $D$ such that every vertex not contained in $D$ has at least $k$ neighbours in $D$. Let $mi_k(n)$ denote the maximum number of $k$-dominating independent sets in a graph on $n$ vertices, and let $ζ_k:=\lim_{n \rightarrow \infty} \sqrt[n]{mi_k(n)}$. Nagy initiated the study of $mi_k(n)$. In this artic…

关于弹性超图

On resilient hypergraphs

本文研究弹性超图的匹配数，猜想完全3-图在给定顶点数下具有最大弹性，并证明了s=2的情形，同时改进了s≥3时的已知上界。

On Gallai's conjecture for series-parallel graphs and planar 3-trees

A path cover is a decomposition of the edges of a graph into edge-disjoint simple paths. Gallai conjectured that every connected $n$-vertex graph has a path cover with at most $\lceil n/2 \rceil$ paths. We prove Gallai's conjecture for series-parallel graphs. For the class of planar 3-trees we show how to construct a path cover with at most $\lfloor 5n/8 \rfloor$ paths, which is an improvement over the best previously known bound of $\lfloor 2n/3 \rfloor$.

The Elekes-Szabó Problem and the Uniformity Conjecture

In this paper we give a conditional improvement to the Elekes-Szabó problem over the rationals, assuming the Uniformity Conjecture. Our main result states that for $F\in \mathbb{Q}[x,y,z]$ belonging to a particular family of polynomials, and any finite sets $A, B, C \subset \mathbb Q$ with $|A|=|B|=|C|=n$, we have \[ |Z(F) \cap (A\times B \times C)| \ll n^{2-\frac{1}{s}}. \] The value of the integer $s$ is dependent on the polynomial $F$, but is always bounded by $s \leq 5$, and so even in the worst applicable case this gives a quantitative improvement on a bound of Raz, Sharir and de Zeeuw (a…

A new bound on partial sum-sets and difference-sets, and applications to the Kakeya conjecture

Let $A, B$, be finite subsets of an abelian group, and let $G \subset A \times B$ be such that $# A, # B, # \{a+b: (a,b) \in G \} \leq N$. We consider the question of estimating the quantity $# \{a-b: (a,b) \in G \}$. Recently Bourgain improved the trivial upper bound of $N^2$ to $N^{2-1/13}$, and applied this to the Kakeya conjecture. We improve Bourgain's estimate further to $N^{2-1/6}$, and obtain the further improvement of $N^{2-1/4}$ if we also know that $# \{a+2b: (a,b) \in G\} \leq N$. We conclude that Besicovitch sets in $\R^n$ have Hausdorff dimension at least 6n/11+5/11 and Minkowski…

An improvement on the Rado bound for the centerline depth

Let $μ$ be a Borel probability measure in $\mathbb R^d$. For a $k$-flat $α$ consider the value $\inf μ(H)$, where $H$ runs through all half-spaces containing $α$. This infimum is called the half-space depth of $α$. Bukh, Matoušek and Nivasch conjectured that for every $μ$ and every $0 \leq k < d$ there exists a $k$-flat with the depth at least $\tfrac{k + 1}{k + d + 1}$. The Rado Centerpoint Theorem implies a lower bound of $\tfrac{1}{d + 1 - k}$ (the Rado bound), which is, in general, much weaker. Whenever the Rado bound coincides with the bound conjectured by Bukh, Matoušek and Nivasch, i.e.…

One more proof of the first linear programming bound for binary codes and two conjectures

We give one more proof of the first linear programming bound for binary codes, following the line of work initiated by Friedman and Tillich. The new argument is somewhat similar to previous proofs, but we believe it to be both simpler and more intuitive. Moreover, it provides the following 'geometric' explanation for the bound. A binary code with minimal distance $δn$ is small because the projections of the characteristic functions of its elements on the subspace spanned by the Walsh-Fourier characters of weight up to $\left(\frac 12 - \sqrt{δ(1-δ)}\right) \cdot n$ are essentially independent.…

使用 decoupling 定理与 two-ends Furstenberg 不等式研究限制性估计

Restriction estimates using decoupling theorems and two-ends Furstenberg inequalities

本文提出利用 decoupling 定理与 two-ends Furstenberg 不等式研究限制性猜想，并给出一个蕴含该猜想的 two-ends Furstenberg 猜想，在平面上予以证明，进而推导出三维 $p>22/7$ 的限制性估计及高维改进。