On symmetric intersecting families of vectors

Abstract

A family of vectors A ⊂ [k]n is said to be intersecting if any two elements of A agree on at least one coordinate. We prove, for fixed k ≥ 3, that the size of a symmetric intersecting subfamily of [k] n is o(k n ), which is in stark contrast to the case of the Boolean hypercube (where k = 2). Our main contribution addresses limitations of existing technology: while there is now spectral machinery, developed by Ellis and the third author, to tackle extremal problems in set theory involving symmetry, this machinery relies crucially on the interplay between up-sets, biased product measures, and threshold behaviour in the Boolean hypercube, features that are notably absent in the problem at hand. To circumvent these barriers, we introduce the notion of a 'measure flow' on the product of 'connected posets', and prove a sharp threshold theorem for such flows, which may be of independent interest.

Forbidden intersections for codes

“…As mentioned above, Theorem 1.2 implies the result of [5], as a junta is far from being symmetric. The assumption 𝑚 ⩾ 3 is necessary, as when 𝑚 = 2, we have symmetric examples as mentioned above.…”

“…Our proof (discussed in the next subsection) proceeds via a junta approximation result of independent interest, showing that any (𝑡 − 1)-avoiding code is approximately contained in a 𝑡-intersecting junta (a code where membership is determined by a constant number of coordinates). In particular, when 𝑡 = 1, this gives an alternative proof of the result of [5], as a family that essentially depends on few coordinates is very far from being symmetric.…”

“…However, these settings are quite different, in that there are many maximum intersecting families of sets, including very symmetric examples such as the family of all sets of size > 𝑛∕2, whereas in [𝑚] 𝑛 for 𝑚 > 2, the only example is obtained by fixing one coordinate to have a fixed value. A more substantial difference was recently demonstrated by Eberhard, Kahn, Narayanan and Spirkl [5], who showed that adding a symmetry assumption reduces the maximum size to 𝑜(𝑚 𝑛 ).…”

Forbidden intersections for codes

“…As mentioned above, Theorem 1.2 implies the result of [5], as a junta is far from being symmetric. The assumption m ≥ 3 is necessary, as when m = 2 we have symmetric examples as mentioned above.…”

“…However, these settings are quite different, in that there are many maximum intersecting families of sets, including very symmetric examples such as the family of all sets of size > n/2, whereas in [m] n for m > 2 the only example is obtained by fixing one coordinate to have a fixed value. A more substantial difference was recently demonstrated by Eberhard, Kahn, Narayanan and Spirkl [5], who showed that adding a symmetry assumption reduces the maximum size to o(m n ).…”