Please help me to find a proper reference to the following infinite version of the Sunflower Lemma.

Lemma. Let $n\in\mathbb N$. Every infinite family of $n$-element sets contains an infinite subfamily $\mathcal F$ such that $A\cap B=\bigcap\mathcal F$ for any distinct sets $A,B\in\mathcal F$.

Browsing through Internet I could find only finite and uncountable versions of the Sunflower Lemma (but not the countable one).

  • $\begingroup$ The uncountable version is usually known (as I'm sure you know, but didn't mention) as the $\Delta$-system lemma. $\endgroup$ – Joel David Hamkins Jul 19 '18 at 21:16
  • $\begingroup$ @JoelDavidHamkins You are right. But the finite Sunflower lemma also often is referred as $\Delta$-lemma, see michaelnielsen.org/polymath1/… $\endgroup$ – Taras Banakh Jul 19 '18 at 21:20
  • $\begingroup$ Indeed, I like the sunflower terminology, which is attractively evocative of the essence of the situation. Is there a common usage of the sunflower terminology in the uncountable case? I've not heard of this terminology until now. I find the sunflower metaphor a little more accurate than the $\Delta$-metaphor. $\endgroup$ – Joel David Hamkins Jul 19 '18 at 21:26
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    $\begingroup$ I had seen the "sunflower" terminology once before. I think it was in a computer science paper. The sunflower picture nicely matches my mental picture of the basic situation. Often, though, I need (or at least want) the improved version that says the root (or kernel or...) is an initial segment of each of the sets in the $\Delta$-system; then the sunflower looks rather lopsided. $\endgroup$ – Andreas Blass Jul 19 '18 at 22:40
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    $\begingroup$ @AndreasBlass: Not at all -- that version of the lemma simply describes freshly opened sunflowers photos.capturememphis.com/photos/1BD_TvOSPqkevXxpdVS3tg/…. $\endgroup$ – Will Brian Jul 20 '18 at 12:53

I found it in the book Komjáth, Péter; Totik, Vilmos, Problems and theorems in classical set theory, Problem Books in Mathematics. New York, NY: Springer (ISBN 0-387-30293-X/hbk). xii, 514 p. (2006). ZBL1103.03041. It's stated on p. 107 as the first item in the chapter on $\Delta$-systems:

  1. An infinite family of $n$-element sets ($n\lt\omega$) includes an infinite $\Delta$-subfamily.

A proof is given on p. 421. No reference is given.


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