Moonflowers and efficient code sparsification

TL;DR

This paper introduces moonflowers, a new combinatorial structure, and applies it to improve bounds in code sparsification, achieving near-optimal results with a logarithmic dependence on block length.

Contribution

We define moonflowers, analyze their extremal properties, and apply these insights to significantly improve bounds in code sparsification problems.

Findings

Established near-optimal bounds for moonflower families avoiding a k-moonflower.

Improved the dependence on block length from poly-logarithmic to logarithmic in code sparsification.

Proved that the logarithmic dependence on block length is necessary.

Abstract

We introduce \emph{moonflowers}, a weaker analogue of sunflowers. A family of sets $S_1,\ldots,S_k$ is a $k$-moonflower if each set $S_i$ contains at least one element that is absent from all the others. We study the extremal problem of determining the largest possible size of a family of sets of size at most $w$ that avoids a $k$-moonflower, and obtain near-optimal bounds. As an application, we revisit the code sparsification problem studied by Brakensiek and Guruswami (STOC 2025) and improve the bounds to near optimal. Concretely, we improve the dependence on the block length from poly-logarithmic to logarithmic, and show that such a dependence is necessary.