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https://doi.org/10.1007/s00493-024-00115-1

Nešetřil, Jaroslav; Rödl, Vojtěch; Sales, Marcelo (July 2024, Combinatorica)

Abstract For any integer$$h\geqslant 2$$

h ⩾ 2

, a set of integers$$B=\{b_i\}_{i\in I}$$

B = {b_{i}}_{i \in I}

is a$$B_h$$

B_{h}

-set if allh-sums$$b_{i_1}+\ldots +b_{i_h}$$

b_{i_{1}} + \dots + b_{i_{h}}

with$$i_1<\ldots

i_{1} < \dots < i_{h}

are distinct. Answering a question of Alon and Erdős [2], for every$$h\geqslant 2$$

h ⩾ 2

we construct a set of integersXwhich is not a union of finitely many$$B_h$$

B_{h}

-sets, yet any finite subset$$Y\subseteq X$$

Y \subseteq X

contains an$$B_h$$

B_{h}

-setZwith$$|Z|\geqslant \varepsilon |Y|$$

| Z | ⩾ ε | Y |

, where$$\varepsilon :=\varepsilon (h)$$

ε : = ε (h)

. We also discuss questions related to a problem of Pisier about the existence of a setAwith similar properties when replacing$$B_h$$

B_{h}

-sets by the requirement that all finite sums$$\sum _{j\in J}b_j$$

\sum_{j \in J} b_{j}

are distinct.

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