Aymen Ben Amira, Jamel Dammak, Hamza Si Kaddour
Kelly's combinatorial lemma is a basic tool in the study of Ulam's reconstruction conjecture. A generalization in terms of a family of t-elements subsets of a v-element set was given by Pouzet. We consider a version of this generalization modulo a prime p. We give illustrations to graphs and tournaments.
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A refinement of Kelly's lemma for graph reconstruction for counting rooted subgraphs
本文精细化Kelly引理，研究计数根顶点与删除顶点重合的有根子图，证明一般不可能，但半径大于k时可计数固定高度k的多重集，并推广至边重构。
The Generalized Combinatorial Lason-Alon-Zippel-Schwartz Nullstellensatz Lemma
本文综述了Alon组合零点定理与Schwartz-Zippel引理的若干强化与推广，证明了当变量在足够大范围内独立取值时多元多项式非零值的存在性。
Green [Geometric and Functional Analysis 15 (2005), 340--376] established a version of the Szemerédi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his Removal Lemma that allows us to extend its statement to all finite groups. We also discuss possible extensions of the Removal Lemma to systems of equations.
Fan's lemma via bistellar moves
本文将 Pachner 的双星形移动定理推广到带自由 Z2 作用的组合流形，并据此给出 Fan 引理的组合证明，且在三维及以下无需额外三角剖分假设。
We develop a geometric framework that unifies several different combinatorial fixed-point theorems related to Tucker's lemma and Sperner's lemma, showing them to be different geometric manifestations of the same topological phenomena. In doing so, we obtain (1) new Tucker-like and Sperner-like fixed-point theorems involving an exponential-sized label set; (2) a generalization of Fan's parity proof of Tucker's Lemma to a much broader class of label sets; and (3) direct proofs of several Sperner-like lemmas from Tucker's lemma via explicit geometric embeddings, without the need for topological f…
Tucker's Lemma is a combinatorial analog of the Borsuk-Ulam theorem and the case n=2 was proposed by Tucker in 1945. Numerous generalizations and applications of the Lemma have appeared since then. In 2006 Meunier proved the Lemma in its full generality in his Ph.D. thesis. There are generalizations and extensions of the Borsuk-Ulam theorem that do not yet have combinatorial analogs. In this note, we give a combinatorial analog of a result of Freeman J. Dyson and show that our result is equivalent to Dyson's theorem. As with Tucker's Lemma, we hope that this will lead to generalizations and ap…
Combinatorial degree version of a generalized $\mathbb{Z}_p$-Tucker's lemma with a combinatorial proof
本文提出了广义Z_p-图克勒引理的组合度版本，并通过纯粹组合的方法证明了该定理，同时引入了一个独立的组合霍普夫追踪公式。
We provide a combinatorial interpretation of Lah numbers by means of planar networks. Henceforth, as a conesquence of Lindström's lemma, we conclude that the related Lah matrix possesses a remarkable property of total non-negativity.
We introduce a combinatorial enumeration problem that is solved using generalized Catalan numbers. We also study generalizations of the Cycle Lemma beyond the computation of the generalized Catalan numbers.
The Springer modules have a combinatorial property called ``coincidence of dimensions,'' i.e., the Springer modules are naturally decomposed into submodules with common dimensions. Morita and Nakajima proved the property by giving modules with common dimensions whose induced modules are isomorphic to the submodules of Springer modules. They proved that the induced modules are isomorphic to the submodules, by showing the coincidence of their characters. Our aim is to construct isomorphisms between the induced modules and their corresponding submodules in a combinatorial manner. For this purpose…
In this work, we present a generalization of Gale's lemma. Using this generalization, we introduce two combinatorial sharp lower bounds for ${\rm conid}({\rm B}_0(G))+1$ and ${\rm conid}({\rm B}(G))+2$, two famous topological lower bounds for the chromatic number of a graph $G$.
Formal Probabilistic Methods for Combinatorial Structures using the Lovász Local Lemma
本文提出在 Isabelle/HOL 中基于 locale 的模块化框架，首次形式化洛瓦兹局部引理，用于形式化组合结构存在性的概率证明，并通过超图着色经典引理验证其适用性。
The Sperner and Tucker lemmas are combinatorial analogous of the Brouwer and Borsuk - Ulam theorems with many useful applications. These classic lemmas are concerning labellings of triangulated discs and spheres. In this paper we show that discs and spheres can be substituted by large classes of manifolds with or without boundary.
We present two proofs each, all shorter than the original proofs, of two elegant combinatorial identities that came up in the beautiful article arXiv:2010.00077 .
The proof of Brouwer's fixed-point theorem based on Sperner's lemma is often presented as an elementary combinatorial alternative to advanced proofs based on algebraic topology. The goal of this note is to show that: (i) the combinatorial proof of Sperner's Lemma can be considered as a cochain-level version, written in the combinatorial language, of a standard cohomological argument; (ii) the standard deduction of Brouwer's theorem from Sperner's lemma is similar to the usual deduction of Brouwer's theorem from the no-retraction theorem and is closely related to the notion of a simplicial appr…
Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rely on a generalization of the OMT and on a lemma about the doubling of manifolds with boundaries that are BUT-manifolds.
We prove an extension of the well-known combinatorial-topological lemma of E. Sperner to the case of infinite-dimensional cubes. It is obtained as a corollary to an infinitary extension of the Lebesgue Covering Dimension Theorem.
A topological proof of Ky Fan's covering lemma
该研究基于Borsuk-Ulam定理给出Ky Fan覆盖引理的直观组合证明，并将该方法推广到多线性序下的Ky Fan覆盖引理。
The present paper is a continuation of author's paper arXiv:1909.00940 [math.AT] devoted to the lemmas of Alexander and Sperner, but is independent from it. We begin by a step back from Alexander and Sperner to Lebesgue work on the invariance of the dimension. In contrast with almost everybody else, Lebesgue worked with cubes rather than with simplices. His methods were developed by Hurewicz and Lusternik-Schnirelmann and then forgotten. In the present paper these methods are recast in the language of cubical chains and cochains. After this, we present a new approach to Lebesgue and Lusternik-…
The Lovász Local Lemma is a very powerful tool in probabilistic combinatorics, that is often used to prove existence of combinatorial objects satisfying certain constraints. Moser and Tardos have shown that the LLL gives more than just pure existence results: there is an effective randomized algorithm that can be used to find a desired object. In order to analyze this algorithm Moser and Tardos developed the so-called entropy compression method. It turned out that one could obtain better combinatorial results by a direct application of the entropy compression method rather than simply appealin…
In view of Tucker's lemma (an equivalent combinatorial version of the Borsuk- Ulam theorem), the present authors (2013) introduced the kth altermatic number of a graph G as a tight lower bound for the chromatic number of G. In this note, we present a purely combinatorial proof for this result.
We give an elementary proof of a generalization of Rokhlin's lemma for commuting non-invertible measure-preserving transformations, and we present several combinatorial applications.
In this paper, we give a new bijective proof of a multiset analogue of even-odd permutations identity. This multiset version is equivalent to the original coin arrangements lemma which is a key combinatorial lemma in the Sherman's Proof of a conjecture of Feynman about an identity on paths in planar graphs related to combinatorial solution of two dimensional Ising model in statistical physics.
We present a constructive proof of Ky Fan's combinatorial lemma concerning labellings of triangulated spheres. Our construction works for triangulations of $S^n$ that contain a flag of hemispheres. As a consequence, we produce a constructive proof of Tucker's lemma that holds for a larger class of triangulations than previous constructive proofs.
The Lovász Local Lemma is a very powerful tool in probabilistic combinatorics, that is often used to prove existence of combinatorial objects satisfying certain constraints. Moser and Tardos have shown that the LLL gives more than just pure existence results: there is an effective randomized algorithm that can be used to find a desired object. In order to analyze this algorithm, Moser and Tardos developed the so-called entropy compression method. It turned out that one could obtain better combinatorial results by a direct application of the entropy compression method rather than simply appeali…
Stiebitz determined the chromatic number of generalised Mycielski graphs using the topological method of Lovasz, which invokes the Borsuk-Ulam theorem. Van Ngoc and Tuza used elementary combinatorial arguments to prove Stiebitz's theorem for 4-chromatic generalised Mycielski graphs, and asked if there is also an elementary combinatorial proof for higher chromatic number. We answer their question by showing that Stiebitz's theorem can be deduced from a version of Fan's combinatorial lemma. Our proof uses topological terminology, but is otherwise completely discrete and could be rewritten to avo…
Local entropy theory, combinatorics, and local theory of Banach spaces
本文比较了群在紧空间及其概率测度空间上作用的局部熵理论，刻画了IE-元组关系，提出新组合引理并应用于Banach空间局部理论。
A group A acting faithfully on a set X is 2-distinguishable if there is a 2-coloring of X that is not preserved by any nonidentity element of A, equivalently, if there is a proper subset of X with trivial setwise stabilizer. The motion of an element a in A is the number of points of X that are moved by a, and the motion of the group A is the minimal motion of its nonidentity elements. For finite A, the Motion Lemma says that if the motion of A is large enough (specifically at least 2 log_2 |A|), then the action is 2-distinguishable. For many situations where X has a combinatorial or algebraic …
An Optimal Sauer Lemma Over $k$-ary Alphabets
本文提出了一种针对多类和列表预测的最优Sauer不等式，通过DS维度和扩展的列表DS维度，改进了多类学习和列表预测的样本复杂度上界。
We discuss five discrete results: the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carathéodory, Helly, and Tverberg from combinatorial geometry. We explore their connections and emphasize their broad impact in application areas such as game theory, graph theory, mathematical optimization, computational geometry, etc.
Motivated by applications in combinatorial geometry, we consider the following question: Let $λ=(λ_1,λ_2,\ldots,λ_m)$ be an $m$-partition of a positive integer $n$, $S_i \subseteq \mathbb{C}^{λ_i}$ be finite sets, and let $S:=S_1 \times S_2 \times \ldots \times S_m \subset \mathbb{C}^n$ be the multi-grid defined by $S_i$. Suppose $p$ is an $n$-variate degree $d$ polynomial. How many zeros does $p$ have on $S$? We first develop a multivariate generalization of Combinatorial Nullstellensatz that certifies existence of a point $t \in S$ so that $p(t) \neq 0$. Then we show that a natural multivari…
Algorithmic mechanism design (AMD) studies the delicate interplay between computational efficiency, truthfulness, and optimality. We focus on AMD's paradigmatic problem: combinatorial auctions. We present a new generalization of the VC dimension to multivalued collections of functions, which encompasses the classical VC dimension, Natarajan dimension, and Steele dimension. We present a corresponding generalization of the Sauer-Shelah Lemma and harness this VC machinery to establish inapproximability results for deterministic truthful mechanisms. Our results essentially unify all inapproximabil…
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