9 pages, 1 figure. Dömötör Pálvölgyi brought to our attention that Kraft's inequality for prefix codes can be used to simplify our argument for the upper bound
One of the most classical results in extremal set theory is Sperner's theorem, which says that the largest antichain in the Boolean lattice $2^{[n]}$ has size $Θ\big(\frac{2^n}{\sqrt{n}}\big)$. Motivated by an old problem of Erdős on the growth of infinite Sidon sequences, in this note we study the growth rate of maximum infinite antichains. Using the well known Kraft's inequality for prefix codes, it is not difficult to show that infinite antichains should be "thinner" than the corresponding finite ones. More precisely, if $\mathcal{F}\subset 2^{\mathbb{N}}$ is an antichain, then
$$\liminf_{n\rightarrow \infty}\big|\mathcal{F} \cap 2^{[n]}\big|\left(\frac{2^n}{n\log n}\right)^{-1}=0.$$ Our main result shows that this bound is essentially tight, that is, we construct an antichain $\mathcal{F}$ such that $$\liminf_{n\rightarrow \infty}\big|\mathcal{F} \cap 2^{[n]}\big|\left(\frac{2^n}{n\log^{C} n}\right)^{-1}>0$$ holds for some absolute constant $C>0$.
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.
It is well known that Sperner lemma is equivalent to Brouwer fixed-point theorem. Tanaka [12] proved that Brouwer theorem is equivalent to Arrow theorem, hence Arrow theorem is equivalent to Sperner lemma. In this paper we will prove this result directly. Moreover, we describe a number of other statements equivalent to Arrow theorem.
Balanced Sperner families via the topological Tverberg theorem
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学交叉科学研究中心）；Frontiers Science Center for Nonlinear Expectations, Ministry of Education（教育部非线性期望前沿科学中心）；School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）
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…
Alexander's lemma is a version of Sperner's lemma published by Alexander two years earlier than Sperner's paper. The present paper is devoted to a modern but elementary exposition of lemmas of Alexander and Sperner and their main topological applications: Brouwer's theorems about the topological invariance of dimension and of domains (here we follow Lebesgue ideas in the form given to them by Sperner), Brouwer's fixed-point theorem, and Alexander's theorem about the topological invariance of homology groups. Along the way we relate the Knaster-Kuratowski-Mazurkiewich argument with the notion o…
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.
For a set $L$ of positive integers, a set system $\mathcal{F} \subseteq 2^{[n]}$ is said to be $L$-close Sperner, if for any pair $F,G$ of distinct sets in $\mathcal{F}$ the skew distance $sd(F,G)=\min\{|F\setminus G|,|G\setminus F|\}$ belongs to $L$. We reprove an extremal result of Boros, Gurvich, and Milani\v c on the maximum size of $L$-close Sperner set systems for $L=\{1\}$ and generalize to $|L|=1$ and obtain slightly weaker bounds for arbitrary $L$. We also consider the problem when $L$ might include 0 and reprove a theorem of Frankl, Füredi, and Pach on the size of largest set systems…
We propose a general technique related to the polytopal Sperner lemma for proving old and new multilabeled versions of Sperner's lemma. A notable application of this technique yields a cake-cutting theorem where the number of players and the number of pieces can be independently chosen. We also prove multilabeled versions of Fan's lemma, a combinatorial analogue of the Borsuk-Ulam theorem, and exhibit applications to fair division and graph coloring.
Sperner theorems for unrelated copies of some partially ordered sets in a powerset lattice and minimum generating sets of powers of distributive lattices
A \emph{chain} in the unit $n$-cube is a set $C\subset [0,1]^n$ such that for every $\mathbf{x}=(x_1,\ldots,x_n)$ and $\mathbf{y}=(y_1,\ldots,y_n)$ in $C$ we either have $x_i\le y_i$ for all $i\in [n]$, or $x_i\ge y_i$ for all $i\in [n]$. We show that the $1$-dimensional Hausdorff measure of a chain in the unit $n$-cube is at most $n$, and that the bound is sharp. Given this result, we consider the problem of maximising the $n$-dimensional Lebesgue measure of a measurable set $A\subset [0,1]^n$ subject to the constraint that it satisfies $\mathcal{H}^1(A\cap C) \le κ$ for all chains $C\subset …
A family of sets is called $r$-\emph{cover free} if no set in the family is contained in the union of $r$ (or less) other sets in the family. A $1$-cover free family is simply an antichain with respect to set inclusion. Thus, Sperner's classical result determines the maximal cardinality of a $1$-cover free family of subsets of an $n$-element set. Estimating the maximal cardinality of an $r$-cover free family of subsets of an $n$-element set for $r>1$ was also studied. In this note we are interested in the following probabilistic variant of this problem. Let $S_0,S_1,\ldots, S_r$ be independent…
Counterexamples to the Balogh-Linz-Patkós Conjecture
School of Mathematics, Nanjing University（南京大学数学系）；School of Mathematics and Statistics, Central China Normal University（华中师范大学数学与统计学院）；Faculty of Environment and Information Sciences, Yokohama National University（横滨国立大学环境信息学部）
In 1967 Herbert Scarf suggested a new proof of Brouwer fixed point theorem based on a surprising analogue of Sperner's lemma. This analogue was motivated by Scarf's work in game theory and mathematical economics. Moreover, Scarf proved a much general version of Sperner's lemma dealing with colorings by vectors. The present paper begins by revisiting Scarf's ideas from the point of view of the basic theory of simplicial cochains in the spirit of author's papers arXiv:1909.00940 and arXiv:2012.13104. After this we get to the main new results of the paper, namely, to a generalization of Scarf res…
In the Boolean lattice, Sperner's, Erdős's, Kleitman's and Samotij's theorems state that families that do not contain many chains must have a very specific layered structure. We show that if instead of $\mathbb{Z}_2^n$ we work in $\mathbb{Z}_{2^n}$, several analogous statements hold if one replaces the word $k$-chain by projective cube of dimension $2^{k-1}$. We say that $B_d$ is a projective cube of dimension $d$ if there are numbers $a_1, a_2, \ldots, a_d$ such that $$B_d = \left\{\sum_{i\in I} a_i \bigg\rvert \emptyset \neq I\subseteq [d]\right\}.$$ As an analog of Sperner's and Erdős's the…
How large an antichain can we find inside a given downset in the lattice of subsets of [n]? Sperner's theorem asserts that the largest antichain in the whole lattice has size the binomial coefficient C(n, n/2); what happens for general downsets? Our main results are a Dilworth-type decomposition theorem for downsets, and a new proof of a result of Engel and Leck that determines the largest possible antichain size over all downsets of a given size. We also prove some related results, such as determining the maximum size of an antichain inside the downset that we conjecture minimizes this quanti…
We give a common matroidal generalisation of `A Cantor-Bernstein theorem for paths in graphs' by Diestel and Thomassen and `A Cantor-Bernstein-type theorem for spanning trees in infinite graphs' by ourselves.
Carmesin has extended Robertson and Seymour's tree-of-tangles theorem to the infinite tangles of locally finite infinite graphs. We extend it further to the infinite tangles of all infinite graphs. Our result has a number of applications for the topology of infinite graphs, such as their end spaces and their compactifications.
A Riemann-Roch theorem on graph was initiated by M. Baker and S. Norine. In their article [2], a Riemann-Roch theorem on a finite graph with uniform vertex-weight and uniform edge-weight was established and it was suggested a Riemann-Roch theorem on an infinite graph was feasible. In this article, we take an edge-weighted infinite graph and focus on the importance of the spectral gaps of the Laplace operators defined on its finite subgraphs naturally given by Q-valued positive weights on the edges. We build a potential theoretic scheme for proof of a Riemann-Roch theorem on the edge-weighted i…
We prove an infinite-dimensional version of an approximate Ramsey theorem of Gowers, initially used to show that every Lipschitz function on the unit sphere of $c_0$ is oscillation stable. To do so, we use the theory of ultra-Ramsey spaces developed by Todorcevic in order to obtain an Ellentuck-type theorem for the space of all infinite block sequences in $\mathrm{FIN}_{\pm k}$.
Building on classical theorems of Sperner and Kruskal-Katona, we investigate antichains $\mathcal F$ in the Boolean lattice $B_n$ of all subsets of $[n]:=\{1,2,\dots,n\}$, where $\mathcal F$ is flat, meaning that it contains sets of at most two consecutive sizes, say $\mathcal F=\mathcal{A}\cup\mathcal{B}$, where $\mathcal{A}$ contains only $k$-subsets, while $\mathcal{B}$ contains only $(k-1)$-subsets. Moreover, we assume $\mathcal{A}$ consists of the first $m$ $k$-subsets in squashed (colexicographic) order, while $\mathcal{B}$ consists of all $(k-1)$-subsets not contained in the subsets in …
We prove a new polynomial refinement of the Capparelli's identities. Using a special case of Bailey's lemma we prove many infinite families of sum-product identities that root from our finite analogues of Capparelli's identities. We also discuss the $q\mapsto 1/q$ duality transformation of the base identities and some related partition theoretic relations.
We show that the infinite-dimensional versions of Gowers' $\mathrm{FIN}_k$ and $\mathrm{FIN}_{\pm k}$ theorems can be parametrized by an infinite sequence of perfect subsets of $2^ω$. To do so, we use ultra-Ramsey theory to obtain exact and approximate versions of a result which combines elements from both Gowers' theorems and the Hales-Jewett theorem. As a consequence, we obtain a parametrized version of Gowers' $c_0$ theorem.
We prove an infinite Ramsey theorem for noncommutative graphs realized as unital self-adjoint subspaces of linear operators acting on an infinite dimensional Hilbert space. Specifically, we prove that if V is such a subspace, then provided there is no obvious obstruction, there is an infinite rank projection P with the property that the compression PVP is either maximal or minimal in a certain natural sense.
