On L-Close Sperner Systems

Abstract

For a set L of positive integers, a set system $${\mathcal F}\subseteq 2^{[n]}$$ F ⊆ 2 [ n ] is said to be L-close Sperner, if for any pair F, G of distinct sets in $${\mathcal F}$$ F the skew distance $$sd(F,G)=\min \{|F\setminus G|,|G\setminus F|\}$$ s d ( F , G ) = min { | F \ G | , | G \ F | } belongs to L. We reprove an extremal result of Boros, Gurvich, and Milanič on the maximum size of L-close Sperner set systems for $$L=\{1\}$$ L = { 1 } , generalize it to $$|L|=1$$ | 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 with all skew distances belonging to $$L=\{0,1\}$$ L = { 0 , 1 } .

Role of Culinary Indian Spices in the Regulation of TGF‐β Signaling Pathway in Inflammation‐Induced Liver Cancer

“…[ 16 ] These immune cells actively protect the liver and eliminate hazardous chemicals, exogenous infections, and their byproducts, early cancer cells and regulate the cancer microenvironment. [ 17 ] Tumor microenvironment contains endothelial cells, inflammatory cells, stromal cells, cytokines (such as IL‐6, IL‐10, TGF β), extracellular matrix, and proangiogenic factor. It also contains microvessels and biomolecules infiltrated in neighboring area plays a crucial role in initiation and development of HCC.…”

Infectious Hospital Area, the First Affiliated Hospital of University of Science and Technology of China, Hefei, China

“…For Sperner families, Nagy and Patkós [20] introduced the notion of L-close Sperner systems. A set system F ⊆ 2 [n] is said to be L-close Sperner, if for any pair of distinct sets A, B in F , the skew distance min{|A \ B|, |B \ A|} ∈ L. Using linear independence arguments, Nagy and Patkós [20] obtained the following result.…”