A unifying generalization of Sperner's theorem

A Proof of Atanassov's Conjecture and Other Generalizations of Sperner's Lemma

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.

Fair division and generalizations of Sperner- and KKM-type results

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…

AZ-identities and Strict 2-part Sperner Properties of Product Posets

One of the central issues in extremal set theory is Sperner's theorem and its generalizations. Among such generalizations is the best-known BLYM inequality and the Ahlswede--Zhang (AZ) identity which surprisingly generalizes the BLYM inequality into an identity. Sperner's theorem and the BLYM inequality has been also generalized to a wide class of posets. Another direction in this research was the study of more part Sperner systems. In this paper we derive AZ type identities for regular posets. We also characterize all maximum 2-part Sperner systems for a wide class of product posets.

Around Sperner's lemma

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.

A Geometric Approach to Combinatorial Fixed-Point Theorems

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…

布尔格与线性格之间LYM不等式的一种关系及其应用

A Relationship for LYM Inequalities between Boolean Lattices and Linear Lattices with Applications

该研究建立了布尔格与线性格间LYM不等式的通用关系，借此将Kleitman定理、Johnston-Lu-Milans定理等多个极值集合论经典定理从布尔格推广到线性格，丰富了Sperner理论的工具与结论。