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Set systems with distinct sumsets

Author
Cilleruelo, J.; Serra, O.; Wötzel, M.
Type of activity
Presentation of work at congresses
Name of edition
Discrete Mathematics Days 2018
Date of publication
2018
Presentation's date
2018-06-27
Book of congress proceedings
Discrete Mathematic Days 2018: Electronic notes in discrete mathematics, Volume 68, July 2018: Sevilla, Spain: june 27-29, 2018: extended abstracts
First page
17
Last page
22
Publisher
Elsevier
DOI
https://doi.org/10.1016/j.endm.2018.06.004 Open in new window
Project funding
Barcelona Graduate School of Mathematics
Discrete, geometric and random structures
Geometric, algebraic and probabilistic combinatorics
Repository
http://hdl.handle.net/2117/124692 Open in new window
URL
https://www.sciencedirect.com/science/article/pii/S1571065318300957?via%3Dihub Open in new window
Abstract
A family $\mathcal{A}$ of $k$-subsets of $\{1,2,\dots, N\}$ is a Sidon system if the sumsets $A+A'$, $A,A'\in \mathcal{A}$ are pairwise distinct. We show that the largest cardinality $F_k(N)$ of a Sidon system of $k$-subsets of $[N]$ satisfies $F_k(N)\le {N-1\choose k-1}+N-k$ and the asymptotic lower bound $F_k(N)=\Omega_k(N^{k-1})$. More precise bounds on $F_k(N)$ are obtained for $k\le 3$. We also obtain the threshold probability for a random system to be Sidon for $k= 2$ and $3$.
Citation
Cilleruelo, J., Serra, O., Wötzel, M. Set systems with distinct sumsets. A: Jornadas de Matemática Discreta y Algorítmica. "Discrete Mathematic Days 2018: Electronic notes in discrete mathematics, Volume 68, July 2018: Sevilla, Spain: june 27-29, 2018: extended abstracts". Elsevier, 2018, p. 17-22.
Keywords
Sidon sets, additive combinatorics, distinct sumsets
Group of research
GAPCOMB - Geometric, Algebraic and Probabilistic Combinatorics

Participants