Combinatorics, Probability and Computing最新文献

On the Ramsey numbers of daisies II 关于雏菊的拉姆齐数 II

Combinatorics, Probability and Computing

Pub Date : 2024-09-18 DOI: 10.1017/s0963548324000208

Marcelo Sales

A $(k+r)$ -uniform hypergraph $H$ on $(k+m)$ vertices is an $(r,m,k)$ -daisy if there exists a partition of the vertices $V(H)=Kcup M$ with $|K|=k$ , $|M|=m$ such that the set of edges of $H$ is all the $(k+r)$ -tuples $Kcup P$ , where $P$ is an <jats:alternatives

如果存在一个顶点分割 $V(H)=Kcup M$，且 $|K|=k$ , $|M|=m$，使得 $H$ 的边集是所有 $(k+r)$ 元组 $Kcup P$，其中 $P$ 是 $M$ 的 $r$ 元组，那么在 $(k+m)$ 顶点上的 $(k+r)$ 均匀超图 $H$ 是一个 $(r,m,k)$ 菊花图。我们得到了一个 $(r-2)$ 的指数迭代下限，即 2$ 颜色的 $(r,m,k)$ 雏菊的拉姆齐数。这与完整 $r$ 图的拉姆齐数的最佳下界的数量级相吻合。

引用次数: 0

List packing number of bounded degree graphs 有界阶数图的包装数列表

Combinatorics, Probability and Computing

Pub Date : 2024-09-18 DOI: 10.1017/s0963548324000191

Stijn Cambie, Wouter Cames van Batenburg, Ewan Davies, Ross J. Kang

We investigate the list packing number of a graph, the least

$k$

such that there are always

$k$

disjoint proper list-colourings whenever we have lists all of size

$k$

associated to the vertices. We are curious how the behaviour of the list packing number contrasts with that of the list chromatic number, particularly in the context of bounded degree graphs. The main question we pursue is whether every graph with maximum degree

$Delta$

has list packing number at most

$Delta +1$

. Our results highlight the subtleties of list packing and the barriers to, for example, pursuing a Brooks’-type theorem for the list packing number.

我们研究了图的列表包装数，即当我们有大小为 $k$ 的列表与顶点相关联时，总有 $k$ 不相交的适当列表着色的最小 $k$。我们很好奇列表包装数的行为与列表色度数的行为有什么不同，尤其是在有界度图的情况下。我们研究的主要问题是，是否每个最大度为 $Delta$ 的图的列表打包数都最多为 $Delta +1$ 。我们的结果凸显了列表打包的微妙之处，以及诸如寻求列表打包数布鲁克斯（Brooks'type）定理的障碍。

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引用次数: 0

A new formula for the determinant and bounds on its tensor and Waring ranks 行列式的新公式及其张量和瓦林等级的界限

Combinatorics, Probability and Computing

Pub Date : 2024-09-18 DOI: 10.1017/s0963548324000233

Robin Houston, Adam P. Goucher, Nathaniel Johnston

We present a new explicit formula for the determinant that contains superexponentially fewer terms than the usual Leibniz formula. As an immediate corollary of our formula, we show that the tensor rank of the

$n times n$

determinant tensor is no larger than the

$n$

-th Bell number, which is much smaller than the previously best-known upper bounds when

$n geq 4$

. Over fields of non-zero characteristic we obtain even tighter upper bounds, and we also slightly improve the known lower bounds. In particular, we show that the

$4 times 4$

determinant over

${mathbb{F}}_2$

has tensor rank exactly equal to

$12$

. Our results also improve upon the best-known upper bound for the Waring rank of the determinant when

$n geq 17$

, and lead to a new family of axis-aligned polytopes that tile

${mathbb{R}}^n$

.

我们为行列式提出了一个新的显式公式，与通常的莱布尼兹公式相比，它包含的项数呈超指数减少。作为我们公式的直接推论，我们证明了 $n times n$ 行列式张量的张量秩不会大于 $n$ -th Bell 数，这比之前已知的当 $n geq 4$ 时的上限要小得多。在非零特征域上，我们得到了更严格的上界，而且还略微改进了已知的下界。特别是，我们证明了 ${mathbb{F}}_2$ 上的 $4 times 4$ 行列式的张量秩正好等于 $12$ 。我们的结果还改进了当 $n geq 17$ 时行列式的瓦林秩的已知上界，并引出了一个新的轴对齐多面体族，它可以平铺 ${mathbb{R}}^n$ 。

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引用次数: 0

On the Ramsey numbers of daisies I 关于拉姆齐雏菊的数量 I

Combinatorics, Probability and Computing

Pub Date : 2024-09-18 DOI: 10.1017/s0963548324000221

Pavel Pudlák, Vojtech Rödl, Marcelo Sales

Daisies are a special type of hypergraph introduced by Bollobás, Leader and Malvenuto. An

$r$

-daisy determined by a pair of disjoint sets

$K$

and

$M$

is the

$(r+|K|)$

-uniform hypergraph

${Kcup P,{:}, Pin M^{(r)}}$

. Bollobás, Leader and Malvenuto initiated the study of Turán type density problems for daisies. This paper deals with Ramsey numbers of daisies, which are natural generalisations of classical Ramsey numbers. We discuss upper and lower bounds for the Ramsey number of

$r$

-daisies and also for special cases where the size of the kernel is bounded.

雏形是波尔洛巴斯、利德和马尔维努托提出的一种特殊类型的超图。由一对不相交的集合 $K$ 和 $M$ 决定的 $r$ -雏形是 $(r+|K|)$ -均匀超图 ${Kcup P,{:}, Pin M^{(r)}}$ 。Bollobás、Leader 和 Malvenuto 发起了对菊花的 Turán 类型密度问题的研究。本文讨论的是雏菊的拉姆齐数，它是经典拉姆齐数的自然概括。我们讨论了 $r$ 雏菊的拉姆齐数的上界和下界，以及内核大小有界的特殊情况。

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引用次数: 0

Counting spanning subgraphs in dense hypergraphs 计算密集超图中的跨越子图

Combinatorics, Probability and Computing

Pub Date : 2024-05-30 DOI: 10.1017/s0963548324000178

Richard Montgomery, Matías Pavez-Signé

We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with a high minimum degree. In particular, for each

$kgeq 2$

and

$1leq ell leq k-1$

, we show that every

$k$

-graph on

$n$

vertices with minimum codegree at least

begin{equation*} left {begin {array}{l@{quad}l} left (dfrac {1}{2}+o(1)right )n & text { if }(k-ell )mid k,[5pt] left (dfrac {1}{lceil frac {k}{k-ell }rceil (k-ell )}+o(1)right )n & text { if }(k-ell )nmid k, end {array} right . end{equation*}

contains

$exp!(nlog n-Theta (n))$

Hamilton

$ell$

-cycles as long as

$(k-ell )mid n$

. When

$(k-ell )mid k$

, this gives a simple proof of a result of Glock, Gould, Joos, Kühn, and Osthus, while when

我们给出了一种简单的方法来估计具有高最小度的超图中某些类别的跨越子图的不同副本的数量。特别是，对于每个 $kgeq 2$ 和 $1leq ell leq k-1$ ，我们证明了在 $n$ 顶点上的每个 $k$ 图的最小度至少是 begin{equation*}.{left {array}{l@{quad}l}left (dfrac {1}{2}+o(1)right )n & text { if }(k-ell )mid k,[5pt] left (dfrac {1}{lceil frac {k}{k-ell }rceil (k-ell )}+o(1)right )n &；text { if }(k-ell )nmid k,end {array}right .end{equation*} 包含 $exp!(nlog n-Theta (n))$ Hamilton $ell$ -cycles as long as $(k-ell )mid n$ .当 $(k-ell )mid k$ 时，这给出了格洛克（Glock）、古尔德（Gould）、乔斯（Joos）、库恩（Kühn）和奥斯特胡斯（Osthus）的一个结果的简单证明，而当 $(k-ell )nmid k$ 时，这给出了一个比费伯（Ferber）、哈迪曼（Hardiman）和蒙德（Mond）给出的，或当 $ell lt k/2$ 时费伯（Ferber）、克里夫列维奇（Krivelevich）和苏达科夫（Sudakov）给出的，或当 $ell lt k/2$ 时，费伯（Ferber）、克里夫列维奇（Krivelevich）和苏达科夫（Sudakov）给出的更弱的计数，但对于一个渐近最优的最小鳕鱼度边界来说，这个计数是成立的。

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引用次数: 0

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