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Abstract Theregularity methodwas pioneered by Szemerédi for graphs and is an important tool in extremal combinatorics. Over the last two decades, several extensions to hypergraphs were developed which were based on seemingly different notions ofquasirandomhypergraphs. We consider the regularity lemmata for three‐uniform hypergraphs of Frankl and Rödl and of Gowers, and present a new proof that the concepts behind these approaches are equivalent.more » « less
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Abstract Daisies are a special type of hypergraph introduced by Bollobás, Leader and Malvenuto. An$$r$$-daisy determined by a pair of disjoint sets$$K$$and$$M$$is the$$(r+|K|)$$-uniform hypergraph$$\{K\cup P\,{:}\, P\in M^{(r)}\}$$. Bollobás, Leader and Malvenuto initiated the study of Turán type density problems for daisies. This paper deals with Ramsey numbers of daisies, which are natural generalisations of classical Ramsey numbers. We discuss upper and lower bounds for the Ramsey number of$$r$$-daisies and also for special cases where the size of the kernel is bounded.more » « less
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Abstract For any integer$$h\geqslant 2$$ , a set of integers$$B=\{b_i\}_{i\in I}$$ is a$$B_h$$ -set if allh-sums$$b_{i_1}+\ldots +b_{i_h}$$ with$$i_1<\ldots are distinct. Answering a question of Alon and Erdős [2], for every$$h\geqslant 2$$ we construct a set of integersXwhich is not a union of finitely many$$B_h$$ -sets, yet any finite subset$$Y\subseteq X$$ contains an$$B_h$$ -setZwith$$|Z|\geqslant \varepsilon |Y|$$ , where$$\varepsilon :=\varepsilon (h)$$ . We also discuss questions related to a problem of Pisier about the existence of a setAwith similar properties when replacing$$B_h$$ -sets by the requirement that all finite sums$$\sum _{j\in J}b_j$$ are distinct.more » « less
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Free, publicly-accessible full text available May 1, 2026
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In their classical paper, Erdős, Goodman and Pósa studied the representation of a graph with vertex set $[n]$ by a family of subsets $$S_1,\dots, S_n$$ with the property that $$\{i,j\}$$ is an edge if and only if $$S_i\cap S_j\neq \emptyset$$. In this note, we consider a similar representation of bounded degree $$r$$-uniform hypergraphs and establish some bounds for a corresponding problem.more » « lessFree, publicly-accessible full text available February 28, 2026
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