Meysam Alishahi, Hossein Hajiabolhassan
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 $G$.
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Generalization and Power of Kocay's Lemma in Graph Reconstruction
该研究推广了Kocay引理并应用于图重构，探讨了推广引理与原引理的效力，证明了树作为图子图的重数可重构，丰富了图重构的理论工具。
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.
A simple proof of Atanassov's Conjecture is presented. Atanassov's Conjecture is a generalization of Sperner's Lemma, a lemma which has been used to prove Brouwer's Fixed Point Theorem, among other fixed point theorems. The proof of Atanassov's Conjecture is based on the Brouwer Degree of maps and is extremely elementary. It is much simpler than the original proofs given for the conjecture and provides some insight into the nature of the conjecture. Furthermore, a generalization of the conjecture is presented and finally a new theorem, similar to the original Sperner Lemma, is proved.
A new generalization of Fiedler's lemma is obtained by introducing the concept of the main function of a matrix. As applications, the universal spectra of the H-join, the spectra of the H-generalized join and the spectra of the generalized corona of any graphs (possibly non-regular) are obtained.
Generalizations of the Crossing Lemma
本文回顾了交叉数领域最重要的不等式Crossing Lemma的若干推广与应用。
The Lovász Local Lemma is a seminal result in probabilistic combinatorics. It gives a sufficient condition on a probability space and a collection of events for the existence of an outcome that simultaneously avoids all of those events. Over the years, more general conditions have been discovered under which the conclusion of the lemma continues to hold. In this note we provide short proofs of two of those more general results: Shearer's lemma and the cluster expansion lemma, in their "lopsided" form. We conclude by using the cluster expansion lemma to prove that the symmetric form of the loca…
A new generalization of Fielder's lemma with applications
本文推广了Fielder引理，统一了已有结果，并用于计算任意图$H$下可交换图$H$-乘积的邻接谱及可交换正则图的通用邻接谱。
Swierczkowski's Lemma - as it is usually formulated - asserts that if f is an at least quaternary operation on a finite set A and every operation obtained from f by identifying a pair of variables is a projection, then f is a semiprojection. We generalize this lemma in various ways. First, it is extended to B-valued functions on A instead of operations on A and to essentially at most unary functions instead of projections. Then we characterize the arity gap of functions of small arities in terms of quasi-arity, which in turn provides a further generalization of Swierczkowski's Lemma. Moreover,…
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 inequality for the Kostka-Foulkes polynomials $K_{λ,μ}(q)$. As a corollary, we obtain a nontrivial lower bound for the Kostka numbers and a new proof of the Berenstein-Zelevinsky weight-multiplicity-one-criterium.
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 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。
In the present paper we propose generalizations of the regularity and counting lemmas for multidimensional matrices under a finite alphabet. Firstly, we prove a variant of a multidimensional regularity lemma with the help of a translation of $\varepsilon$-regularity from graphs to matrices. Next, we state that this $\varepsilon$-regularity is sufficient for obtaining a matrix analogue of the counting lemma for $2$-dimensional matrices but not for higher-dimensional cases. Finally, we introduce $\varepsilon$-regular patterns that allow us to deduce a multidimensional counting lemma.
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…
We treat problems of fair division, their various interconnections, and their relations to Sperner's lemma and the KKM theorem as well as their variants. We prove extensions of Alon's necklace splitting result in certain regimes and relate it to hyperplane mass partitions. We show the existence of fair cake division and rental harmony in the sense of Su even in the absence of full information. Furthermore, we extend Sperner's lemma and the KKM theorem to (colorful) quantitative versions for polytopes and pseudomanifolds. For simplicial polytopes our results turn out to be improvements over the…
A generalization of Croot-Lev-Pach's Lemma and a new upper bound for the size of difference sets in polynomial rings
本文推广Croot-Lev-Pach多项式引理，结合陶哲轩切片秩估计与Gröbner基技术，改进Green的结果，给出多项式环中差集大小及无等差数列集合大小的新上界。
In this paper we consider several generalizations of the Borsuk-Ulam theorem for G-spaces and apply these results to Tucker type lemmas for G-simplicial complexes and PL-manifolds.
We consider a generalization of the classic Sperner lemma. This lemma states that every Sperner coloring of a triangulation of a simplex contains a fully colored simplex. We found a weaker assumption than Sperner's coloring. It is also shown that the main theorem implies Tucker's lemma and some other theorems.
Crossing lemmas for $k$-systems of arcs
该论文推广了可定向曲面上多图的交叉引理，考虑边对为非同伦简单弧且每对至多交叉$k$次的情况。
An elliptic $BC_n$ generalization of the classical two parameter Bailey Lemma is proved, and a basic one parameter $BC_n$ Bailey Lemma is obtained as a limiting case. Several summation and transformation formulas associated with the root system $BC_n$ are proved as applications, including a $_6ϕ_5$ summation formula, a generalized Watson transformation and an unspecialized Rogers--Selberg identity. The last identity is specialized to give an infinite family of multilateral Rogers--Selberg identities. Standard determinant evaluations are then used to compute $B_n$ and $D_n$ generalizations of t…
Computing Non-Repetitive Sequences with a Computable Lefthanded Local Lemma
本文证明了左手洛瓦兹局部引理的可计算版本，并首次构造性地证明了若干类非重复序列的存在性结果。
We consider generalizations of Gale's colored KKM lemma and Shapley's KKMS theorem. It is shown that spaces and covers can be much more general and the boundary KKM rules can be substituted by more weaker boundary assumptions.
We study the complexity of computational problems arising from existence theorems in extremal combinatorics. For some of these problems, a solution is guaranteed to exist based on an iterated application of the Pigeonhole Principle. This results in the definition of a new complexity class within TFNP, which we call PLC (for "polynomial long choice"). PLC includes all of PPP, as well as numerous previously unclassified total problems, including search problems related to Ramsey's theorem, the Sunflower theorem, the Erdős-Ko-Rado lemma, and König's lemma. Whether the first two of these four prob…
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…
The purpose of this article is to formulate conjectural generalizations of Hindman's Theorem and Ellis's Lemma for nonassociative binary systems and relate them to the amenability problem for Thompson's group $F$. Partial results are obtained for both conjectures. The paper will also contain some general analysis of the conjectures.
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