David Conlon, Jacob Fox, Benny Sudakov
We prove several results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are quite short.
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Short proofs of some extremal results
我们证明了几个结果，从不同领域的极值组合，提供完整或部分解决方案的一些开放的问题。这些结果，来自领域，如极值图论，拉姆齐理论和添加剂组合，已收集在一起，因为在每种情况下，有关的证明是相当短的。
We prove a selection of results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are reasonably short.
We determine the maximum number of edges in a $K_4$-minor-free $n$-vertex graph of girth $g$, when $g = 5$ or $g$ is even. We argue that there are many different $n$-vertex extremal graphs, if $n$ is even and $g$ is odd.
Refined Absorption: A New Proof of the Existence Conjecture and its Applications to Extremal and Probabilistic Design Theory
本研究介绍精细吸收法，用其给出组合设计存在性猜想的新证明，该方法可解决极值与概率设计理论的开放问题，提供统一框架，且核心吸收定理可作黑盒使用，免去重复证明。
The following sharpening of Turán's theorem is proved. Let $T_{n,p}$ denote the complete $p$--partite graph of order $n$ having the maximum number of edges. If $G$ is an $n$-vertex $K_{p+1}$-free graph with $e(T_{n,p})-t$ edges then there exists an (at most) $p$-chromatic subgraph $H_0$ such that $e(H_0)\geq e(G)-t$. Using this result we present a concise, contemporary proof (i.e., one applying Szemerédi's regularity lemma) for the classical stability result of Simonovits.
In this paper we estimate the planar Turán number $\mathrm{ex}_\mathcal{P}(n,H)$ of some graphs $H$, i.e., the maximum number of edges in a planar graph $G$ of $n$ vertices not containing $H$ as a subgraph. We give a new, short proof when $H=C_5$, and study the cases when $G$ is bipartite or triangle-free and $H$ is a short even cycle. The proofs are mostly new applications or variants of the "contribution method" introduced by Ghosh, Győri, Martin, Paulos and Xiao in arXiv:2004.14094.
Both the Simonovits stability theorem and the Nikiforov spectral stability theorem are powerful tools for solving exact values of Turán numbers in extremal graph theory. Recently, Füredi [J. Combin. Theory Ser. B 115 (2015)] provided a concise and contemporary proof of the Simonovits stability theorem. In this note, we present a unified treatment for some extremal graph problems, including short proofs of Nikiforov's spectral stability theorem and the clique stability theorem proved recently by Ma and Qiu [European J. Combin. 84 (2020)]. Moreover, some spectral extremal problems related to the…
In this work, new construction methods for self-dual codes are given. The methods use the short Kharaghani array and a variation of it. These are applicable to any commutative Frobenius ring. We apply the constructions over the ring F_2 + uF_2 and self-dual Type I [64, 32, 12]_2-codes with various weight enumerators obtained as Gray images. By the use of an extension theorem for self-dual codes we were able to construct 27 new extremal binary self-dual codes of length 68. The existence of the extremal binary self-dual codes with these weight enumerators was previously unknown.
Let $G$ be a non-bipartite graph which does not contain any odd cycle of length at most $2k+1$. In this paper, we determine the maximum $Q$-index of $G$ if its order is fixed, and the corresponding extremal graph is uniquely characterized. Moreover, if the size of $G$ is given, the maximum $Q$-index of $G$ and the unique extremal graph are also proved.
Extremal Combinatorics is among the most active topics in Discrete Mathematics, dealing with problems that are often motivated by questions in other areas, including Theoretical Computer Science and Information Theory. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers. The topics considered here include questions in Extremal Graph Theory, Coding Theory and Social Choice. This is by no means a comprehensive survey of the area, and is merely a collection of problems, results and proofs,…
Extremal problems for ordered hypergraphs: small patterns and some enumeration
本文研究了不含固定超图F的n阶简单超图H的最大边数和最大点-边关联数的极值函数ex_e（F，n）和ex_i（F，n），其包含性与顶点的线性序有关.我们确定这两个功能，如果F只有不同的单点边缘，或者如果F是一个55超图，最多四个发病率（我们只证明了六种情况）。本文证明了具有n个关联的简单超图和全超图的数的一些精确公式和递推公式，并得到了这些数的粗糙对数渐近性。给出了类似于贝尔数的多宾斯基公式的恒等式。
An Introduction to Razborov's Flag Algebra as a Proof System for Extremal Graph Theory
本文介绍了Razborov旗帜代数在极值图论中的应用，通过逻辑视角阐述其语法、语义和证明策略，并通过实例展示如何利用该框架进行数学论证。
An edge-coloring of a connected graph $G$ is called a {\em monochromatic connection coloring} (MC-coloring for short) if any two vertices of $G$ are connected by a monochromatic path in $G$. For a connected graph $G$, the {\em monochromatic connection number} (MC-number for short) of $G$, denoted by $mc(G)$, is the maximum number of colors that ensure $G$ has a monochromatic connection coloring by using this number of colors. This concept was introduced by Caro and Yuster in 2011. They proved that $mc(G)\leq m-n+k$ if $G$ is not a $k$-connected graph. In this paper we depict all graphs with $m…
The Postage Stamp Problem - Formulae and proof for the case of 3 denominations
A_k = {1，a_2，.，a_k}是n的h-基，如果每个不超过n的正整数都可以表示为不超过h个值a_i的和;极值h-基A_k是n尽可能大的基。计算这样的极值基已成为众所周知的邮票问题。在1968年格尔德霍夫迈斯特出版公式的解决方案邮票问题的三个面额（h = 3）。本文提出了一种替代方法独立开发的作者25年后。
We prove that if $A \subseteq [X, 2X]$ and $B \subseteq [Y, 2Y]$ are sets of integers such that $\gcd(a,b) \geq D$ for at least $δ|A||B|$ pairs $(a,b) \in A \times B$ then $|A||B| \ll_{\varepsilon} δ^{-2 - \varepsilon} XY/D^2$. This is a new result even when $δ= 1$. The proof uses ideas of Koukoulopoulos and Maynard and some additional combinatorial arguments.
We prove a 2018 conjecture of Krawchuk and Rampersad on the extremal behavior of $c(n)$, where $c(n)$ counts the number of length-$n$ factors of the Thue-Morse word $\mathbf{t}$, up to cyclic rotation.
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