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Statement

Setting (p. 2). Put $r(p,k)=\alpha p\log(pk)$, with $\alpha$ the constant the paper chooses. A sequence $S_1,\ldots,S_\ell\subset[n]$ of sets of size $k$ is $r$-spread if, for every nonempty $Z\subset[n]$, at most $r^{k-|Z|}$ elements of the sequence contain $Z$. The paper states its results for sequences, which may repeat sets (footnote 2, p. 2), and notes that a similar notion was first used by Talagrand (footnote 1, p. 2).

Lemma 2 (p. 2, quoted). "If a sequence of more than $r(p,k)^k$ sets of size $k$ is $r(p,k)$-spread, then the sequence must contain $p$ disjoint sets."

The paper proves it "for an appropriate choice of $\alpha$" (p. 2); the logarithm is to base 2 (p. 3).

The sunflower conjecture remark (p. 2). The paper says that, as far as it knows, Lemma 2 may hold even with $r(p,k)=O(p)$, and that such a strengthening would imply the sunflower conjecture of Erdős and Rado.

Source. Anup Rao, Coding for sunflowers, Discrete Analysis 2020:2, 8 pp., doi:10.19086/da.11887 (arXiv:1909.04774v2). Lemma 2 and the remark are on p. 2, the proof on pp. 2--3. Card: Rao 2020.

Read depth. Claims checked: the definition of an $r$-spread sequence, the statement and the remark were read clause by clause on the printed page. The proof was read for structure only.

Proof pointer

Pages 2--3. Apply Lemma 4 with $\gamma=1/(2p)$ and $\varepsilon=1/p$, so that $r(k,\gamma,\varepsilon)=r(p,k)$. Split $[n]$ uniformly at random into $p$ parts $W_1,\ldots,W_p$, each of size at least $\lfloor n/p\rfloor\ge\gamma n$. By symmetry and linearity of expectation, $\mathbb E\bigl[|\chi(X,W_1)|+\cdots+|\chi(X,W_p)|\bigr]<\varepsilon p=1$, so for some fixed partition the sum vanishes for every $X$; then each $W_i$ contains a set of the sequence, and these $p$ sets are pairwise disjoint. Lemma 4 asks for $\varepsilon<1/2$, which $\varepsilon=1/p$ satisfies when $p\ge3$; the paper does not comment on smaller $p$.

Dependencies

Lemma 4 (p. 2), with Definition 3 (p. 2).

Bears on

• Problem 20: Lemma 2 yields Theorem 1, which does not answer the problem. The paper's remark says that Lemma 2 with $r(p,k)=O(p)$ would imply the Erdős--Rado sunflower conjecture, which is the affirmative answer to the problem's question; the paper does not prove that strengthening.