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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.
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Small subsets with large sumset: Beyond the Cauchy–Davenport bound
Abstract For a subset$$A$$of an abelian group$$G$$, given its size$$|A|$$, its doubling$$\kappa =|A+A|/|A|$$, and a parameter$$s$$which is small compared to$$|A|$$, we study the size of the largest sumset$$A+A'$$that can be guaranteed for a subset$$A'$$of$$A$$of size at most$$s$$. We show that a subset$$A'\subseteq A$$of size at most$$s$$can be found so that$$|A+A'| = \Omega (\!\min\! (\kappa ^{1/3},s)|A|)$$. Thus, a sumset significantly larger than the Cauchy–Davenport bound can be guaranteed by a bounded size subset assuming that the doubling$$\kappa$$is large. Building up on the same ideas, we resolve a conjecture of Bollobás, Leader and Tiba that for subsets$$A,B$$of$$\mathbb{F}_p$$of size at most$$\alpha p$$for an appropriate constant$$\alpha \gt 0$$, one only needs three elements$$b_1,b_2,b_3\in B$$to guarantee$$|A+\{b_1,b_2,b_3\}|\ge |A|+|B|-1$$. Allowing the use of larger subsets$$A'$$, we show that for sets$$A$$of bounded doubling, one only needs a subset$$A'$$with$$o(|A|)$$elements to guarantee that$$A+A'=A+A$$. We also address another conjecture and a question raised by Bollobás, Leader and Tiba on high-dimensional analogues and sets whose sumset cannot be saturated by a bounded size subset.
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- PAR ID:
- 10644753
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Combinatorics, Probability and Computing
- Volume:
- 33
- Issue:
- 4
- ISSN:
- 0963-5483
- Page Range / eLocation ID:
- 411 to 431
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/S0963548324000014
