This could really use some examples.<p>So the main gist is: "a collection of sets whose pairwise intersection is constant."<p>So 100 sets that all have different elements would be a sunflower since their intersection is constant (empty)? And 100 sets that all contain some unique elements and some shared across all would be a sunflower, but as soon as some sets share an element that a third doesn't, then not? And two sets would _always_ be a sunflower?<p>Or is the meaning of "constant pairwise intersection" something else?
I just fixed the somewhat confusing wording in the introduction which talked about a constant intersection.<p>I believe the right concept is that the intersection is <i>common</i>.<p>If a set of sets have pairwise intersections that are different, those are still constants, if the sets being considered are constants! E.g. { 1, 2, 3, 4, 5 } is a constant set, so is the { 2 } which is an intersection of { 1, 2 } and { 2, 3 }, and so is { 4 }, which is an intersection of { 3, 4 } and { 1, 4 }.<p>(I get that the idea is that the intersection constant from pair to pair: i.e. if we enumerate the pairs as an index i from 0 to n-1, then the intersection doesn't vary with i.)<p>Under the formal definition, I replaced the explanatory ("in other words") text with a different idea, instead of reiterating what's already in the introduction.<p>Every element in U is either common to all the subsets in W, or else is found in at most one W subset. No elements are shared by some sets in W, but not others.
The existence of sunflowers in any sufficiently large set is an example of a fascinating field of combinatorics called Ramsey theory. The idea is that for a very wide range of structures, sufficiently large objects always necessarily contain substructures of arbitrary size because there's no way to avoid it as the amount of space inside the object grows.<p>The classic example is the "Theorem on Friends and Strangers", which states that in any group of six or more people, there is always some subset of three people who are either pairwise friends or pairwise strangers (or, in graph theoretic terms, three vertices that are either all pairwise connected by an edge or pairwise not connected by an edge).<p>This generalizes to larger subsets: in any group of 18 or more people, there's always some subset of <i>four</i> people who are pairwise friends or strangers, and in any group of 49 or more, there's always some subset of five (49 may not be optimal here; it's known to be between 43 and 49). The known bounds are quite weak: the number of people required to guarantee a mutual-friends-or-strangers subset of <i>n</i> people is known only to be omega(2^(n/2)) and little-o(2^2n), with no improvements on those bounds made despite nearly a century of work.