关于Hendry猜想的新结果

傅里叶熵-影响猜想的进一步证据

Further evidence towards the Fourier Entropy-Influence conjecture

本文通过引入关键不等式和停止二叉树概念，证明了若干新布尔函数类满足傅里叶熵-影响猜想。

关于“第二个”Kahn--Kalai猜想：团、圈与树

On the "second" Kahn--Kalai Conjecture: cliques, cycles, and trees

本文证明了蕴含第二个Kahn--Kalai猜想的随机图陈述的几个简单情形，并希望所引入的思想能推动该自然问题的进一步研究。

关于copartition乘积非负性猜想的进一步结果

Further results on non-negativity conjectures for copartition products

本文针对copartition乘积的有限和无限系数非负性猜想，通过分解为局部块、证明特定情形及改进非负范围，取得了多项新进展。

Further Progress towards the List and Odd Versions of Hadwiger's Conjecture

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Recently, Norin, Song and the author showed that every graph with no $K_t$ minor is $O(t(\log t)^β)$-colorable for every $β> 1/4$, making the first improvement on the order of magnitude of the $O(t\sqrt{\log t})$ bound. Building on that work, we previously showed that every graph with no $K_t$ minor is $O(t (\log…

A conjectured bound on the spanning tree number of bipartite graphs

The Ferrers bound conjecture is a natural graph-theoretic extension of the enumeration of spanning trees for Ferrers graphs. We document the current status of the conjecture and provide a further conjecture which implies it.

New conjectures on algebraic connectivity and the Laplacian spread of graphs

We conjecture a new lower bound on the algebraic connectivity of a graph that involves the number of vertices of high eccentricity in a graph. We prove that this lower bound implies a strengthening of the Laplacian Spread Conjecture. We discuss further conjectures, also strengthening the Laplacian Spread Conjecture, that include a conjecture for simple graphs and a conjecture for weighted graphs.

关于Wojda猜想的进一步进展

Further progress on Wojda's conjecture

本文证明了当m≥93且n≥31m时，Wojda猜想成立，即两个特定大小的有向图可以打包。

Aharoni 对 Caccetta-Häggkvist 猜想的彩虹推广的进一步逼近

Further approximations for Aharoni's rainbow generalization of the Caccetta-Häggkvist conjecture

本文研究 Aharoni 对 Caccetta-Häggkvist 猜想的彩虹推广，将每个颜色类大小的下界从 Ω(k log k) 改进为 Ω(k)，证明此时仍存在长度至多 ⌈n/k⌉ 的彩虹环，并进一步讨论了颜色数量条件放宽后的情形。

Some results on Quillen's Conjecture via equivalent-poset techniques

We extend the Main Theorem of Aschbacher and Smith on Quillen's Conjecture from $p>5$ to the remaining odd primes $p = 3,5$. In the process, we develop further combinatorial and homotopical methods for studying the poset of nontrivial elementary abelian $p$-subgroups of a finite group. The techniques lead to a number of further results on the Conjecture, often reducing dependence on the CFSG; in particular, we also provide some partial results toward the case of $p=2$.

关于并闭集族第二频率的进一步分析

Further analysis on the second frequency of union-closed set families

本文针对 Nagel 关于并闭集族元素频率的强化猜想，利用线性规划进一步刻画 k=2 时集族不满足该猜想的情形。

Counterexamples to the interpolating conjecture on partial-dual genus polynomials of ribbon graphs

Gross, Mansour and Tucker introduced the partial-dual orientable genus polynomial and the partial-dual Euler genus polynomial. They showed that the partial-dual genus polynomial for an orientable ribbon graph is interpolating and gave an analogous conjecture: The partial-dual Euler-genus polynomial for any non-orientable ribbon graph is interpolating. In this paper, we first give some counterexamples to the conjecture. Then motivated by these counterexamples, we further find two infinite classes of counterexamples.

图补猜想弱版本及Delta猜想的部分结果

The Weak Version of the Graph Complement Conjecture and Partial Results for the Delta Conjecture

本文利用Mader经典结果建立了图补猜想的弱版本，提出了$δ$-猜想的极值解，结合图退化改进了界限，并列出了Colin de Verdière数半正定变体的猜想界限。

Solving a conjecture about tessellation graphs of $\mathbb R^2$

In the paper Planarity and Hyperbolicity in Graphs, the authors present the following conjecture: every tessellation of the Euclidean plane with convex tiles induces a non-hyperbolic graph. It is natural to think that this statement holds since the Euclidean plane is non-hyperbolic. Furthermore, there are several results supporting this conjecture. However, this work shows that the conjecture is false.

On Kalai's conjectures concerning centrally symmetric polytopes

In 1989 Kalai stated the three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known as the ``$3^d$-conjecture''. It is well-known that the three conjectures hold in dimensions d \leq 3. We show that in dimension 4 only conjectures A and B are valid, while conjecture C fails. Furthermore, we show that both conjectures B and C fail in all dimensions d \geq 5.

Further $q$-supercongruences from a transformation of Rahman

Employing a quadratic transformation formula of Rahman and the method of `creative microscoping' (introduced by the author and Zudilin in 2019), we provide some new $q$-supercongruences for truncated basic hypergeometric series. In particular, we confirm two recent conjectures of Liu and Wang. We also propose some related conjectures on supercongruences and $q$-supercongruences.

On the Further Structure of the Finite Free Convolutions

Since the celebrated resolution of Kadison-Singer (via the Paving Conjecture) by Marcus, Spielman, and Srivastava, much study has been devoted to further understanding and generalizing the techniques of their proof. Specifically, their barrier method was crucial to achieving the required polynomial root bounds on the finite free convolution. But unfortunately this method required individual analysis for each usage, and the existence of a larger encapsulating framework is an important open question. In this paper, we make steps toward such a framework by generalizing their root bound to all dif…