On symmetric intersecting families of vectors

Abstract

A family of vectors in [k] n is said to be intersecting if any two of its elements agree on at least one coordinate. We prove, for fixed k ≥ 3, that the size of any intersecting subfamily of [k] n invariant under a transitive group of symmetries 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 are now methods, first appearing in work of Ellis and the third author, for using spectral machinery to tackle problems in extremal 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 considered here. To circumvent these barriers, introducing ideas that seem of independent interest, we develop a variant of the sharp threshold machinery that applies at the level of products of posets.

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 $$m \geqslant 3$$ 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 $$o(m^n)$$.…”

On $t$-Intersecting Families of Permutations

“…Ihringer and Kupavskii [27] obtained results in this spirit for regular families of k-element subsets of [n] (i.e., families in which each element is included in the same number of sets); those results were recently quantitatively improved by Kupavskii and Zakharov [33]. Eberhard, Kahn, Narayanan, and Spirkl [7] obtained similar results for symmetric intersecting families of vectors in [m] n .…”