Erdős Problem #328

Suppose $A\subseteq\mathbb{N}$ and $C>0$ is such that $1_A\ast 1_A(n)\leq C$ for all $n\in\mathbb{N}$. Can $A$ be partitioned into $t$ many subsets $A_1,\ldots,A_t$ (where $t=t(C)$ depends only on $C$) such that $1_{A_i}\ast 1_{A_i}(n)<C$ for all $1\leq i\leq t$ and $n\in \mathbb{N}$?

#328: [ErGr80,p.48]

number theory | additive combinatorics

Asked by Erdős and Newman. Nešetřil and Rödl [NeRo85] have shown the answer is no for all $C$ (even if $t$ is also allowed to depend on $A$).

Erdős [Er80e] had previously shown the answer is no for $C=3,4$ and infinitely many other values of $C$.

See also [774].

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This page was last edited 28 December 2025.

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Additional thanks to: Desmond Weisenberg

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