Moonflowers and efficient code sparsification

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.

• The paper introduces moonflowers, a relaxation of sunflowers equivalent to induced matchings in bipartite graphs, and proves near-optimal bounds of $(Ck/w)^w$ for $w\le k$ and $(Cw/k)^k$ for $w\ge k$.
• The authors combine entropy bounds, LP-duality-based coverability, and iterative puncturing to control moonflower-free set systems, achieving tight results up to constants in the exponential base.
• The paper applies these bounds to code sparsification, improving prior $(\log n)^6$ dependence to a single necessary $\log n$ factor up to polylogarithmic terms, while leaving the optimal dependence on $\varepsilon$ open.

Overview

This paper, by Lovett, Meka, and Wang (2605.08676), introduces moonflowers, a relaxation of sunflowers in extremal set theory, and proves near-optimal bounds for the associated extremal problem. The main combinatorial result states that a family of $w$-sets containing no $k$-moonflower has size at most $(Ck/w)^w$ when $w \le k$ and at most $(Cw/k)^k$ when $w \ge k$, for an absolute constant $C$. This is tight up to constant factors in the base of the exponent via an explicit construction. As an application, the authors improve the code sparsification bounds of Brakensiek and Guruswami (STOC 2025), reducing the dependence on block length $n$ from polylogarithmic to a single logarithmic factor and proving that this dependence is necessary.

Moonflowers: definition and relation to sunflowers

A family of sets $S_1,\dots,S_k$ is a $k$-moonflower if there exists a set $k$0 (the core) such that the differences $k$1 are nonempty and pairwise disjoint; no condition is imposed on the intersections $k$2. Every sunflower is either a moonflower or becomes one after deleting one petal, but not conversely: adding a fresh element to each set of an arbitrary set system yields a moonflower that is generally not a sunflower. The gap can be exponential — the paper exhibits a family of size $k$3 with no 3-sunflower whose augmentation by private elements forms a $k$4-moonflower.

The equivalence to induced matchings in bipartite graphs is immediate: viewing a set family as a bipartite graph between sets and support elements, $k$5-moonflowers correspond exactly to induced matchings of size $k$6. Consequently, the main theorem yields a matching extremal bound on left-degree-$k$7 bipartite graphs without large induced matchings.

Extremal bound and proof technique

The upper bound proof follows the high-level architecture of Brakensiek–Guruswami but achieves quantitative improvements through three modular steps:

1. Entropy bound: any $k$8-smooth distribution supported on a $k$9-moonflower-free family has entropy at most $(Ck/w)^w$0, obtained by combining the fact that moonflower-freeness implies $(Ck/w)^w$1 for the union closure, Sauer–Shelah, and Gilmer/Sawin entropy amplification.
2. Coverability from smoothness: via LP duality (von Neumann's minimax theorem) and a clean convexity argument, if all $(Ck/w)^w$2-smooth distributions have entropy at most $(Ck/w)^w$3, then removing at most $(Ck/w)^w$4 sets leaves a $(Ck/w)^w$5-covered family.
3. One-step puncturing: an iterative sampling argument with potential $(Ck/w)^w$6 shows that from a $(Ck/w)^w$7-almost-covered family one can find $(Ck/w)^w$8 of size $(Ck/w)^w$9 covering all but a $w \le k$0 fraction minus $w \le k$1 sets.

Iterating this puncturing yields the theorem. In the $w \le k$2 case, each round shrinks the universe ratio $w \le k$3 under the map $w \le k$4, so only $w \le k$5 rounds are needed while losing at most a constant fraction of the family. The $w \le k$6 case requires reducing the entropy parameter $w \le k$7 down to $w \le k$8 first, which takes $w \le k$9 rounds of halving.

Tightness. The lower bound construction — all $(Cw/k)^k$0-subsets of a universe of size $(Cw/k)^k$1 — gives $(Cw/k)^k$2 sets with no $(Cw/k)^k$3-moonflower, since any $(Cw/k)^k$4-moonflower must span at least $(Cw/k)^k$5 elements. Hence the upper bound is optimal up to constants in the base across all regimes of $(Cw/k)^k$6 and $(Cw/k)^k$7. Notably, when $(Cw/k)^k$8, moonflower-free families have size at most $(Cw/k)^k$9, in contrast to sunflowers where the corresponding bound remains open up to the sunflower conjecture.

Code sparsification

An $w \ge k$0-sparsifier of a code $w \ge k$1 is a weighted coordinate set $w \ge k$2 such that $w \ge k$3 for every codeword. The key structural observation is that non-redundancy equals the largest moonflower in the support family: $w \ge k$4. For linear codes, $w \ge k$5 coincides with dimension, recovering the $w \ge k$6 sparsifiers of Khanna–Putterman–Sudan as a special case.

The main result improves Brakensiek–Guruswami's $w \ge k$7 dependence:

$w \ge k$8

The proof stratifies codeword weights into three regimes. Tiny weights ($w \ge k$9) are captured deterministically, costing at most $C$0 coordinates. Medium weights ($C$1) are handled per dyadic scale using a puncturing lemma that produces coordinate sets of size $C$2 while ensuring traces outside the punctured set number at most $C$3, making Chernoff-plus-union-bound viable. Large weights ($C$4) require no puncturing: the tight moonflower bound caps the number of codewords in a dyadic layer at $C$5, so direct union bounding suffices with per-round error $C$6 that decreases with weight. A recursive "puncture–then-halve" process runs for $C$7 rounds; error accumulation is controlled because residual weights shrink geometrically, giving a convergent series rather than requiring $C$8 — this is precisely what avoids the extra $C$9 losses in prior work.

Lower bound. An explicit construction (a union of $n$0 chains on disjoint ground sets, with geometrically growing prefix lengths) achieves $n$1 yet forces any $n$2-sparsifier to satisfy

$n$3

Thus the single $n$4 dependence in the upper bound is necessary, resolving the optimality question left open by Brakensiek–Guruswami. For constant $n$5, the lower bound matches the upper bound up to polynomial factors in $n$6.

Limitations and open questions

Several caveats bear directly on the results. All results are existential; the authors note that making them algorithmic appears difficult, particularly the step of algorithmically identifying the exceptional set of size $n$7 whose removal renders a family $n$8-covered. The sparsification lower bound is suboptimal in its $n$9 dependence ($S_1,\dots,S_k$0 versus $S_1,\dots,S_k$1 in the upper bound); the authors conjecture $S_1,\dots,S_k$2 is correct but leave this open, as well as matching lower bounds for balanced codes. Finally, given the tight bounds and simple structure of moonflowers, the authors pose the broader question of what other applications exist beyond induced matchings and code sparsification.

Conclusion

The paper establishes moonflowers as a natural sunflower variant admitting essentially tight extremal bounds — a contrast with the still-open sunflower conjecture — and demonstrates their utility by bringing code sparsification to within $S_1,\dots,S_k$3 factors of optimal in the block length, with a matching necessity result for the $S_1,\dots,S_k$4 factor. The remaining gaps concern the $S_1,\dots,S_k$5-dependence of sparsification and the algorithmic realization of the existential arguments.