On a Taylor-Zwicker Construction for Balanced Families and a Conjecture of Moss and Pedersen
2026-06-15 • Discrete Mathematics
Discrete Mathematics
AI summaryⓘ
The authors study collections of equal-sized subsets from a set with 2n elements, focusing on those where each element appears equally often across the collection (called balanced families). They explore how large such families can be without containing small balanced subfamilies of sizes 2, 4, up to 2k. They show that for large enough n, one can build a large balanced family that avoids these smaller balanced groups but contains a balanced subfamily of size 2k+2. Their proof uses advanced combinatorial constructions related to magic-square games, confirming a recent conjecture by Moss and Pedersen.
Boolean latticemiddle layerbalanced familycombinatoricssubset familiescomplementary pairstrade-robustnessmagic-square gamesself-dual selectorsconjecture proof
Authors
Sa'ul A. Blanco
Abstract
We study balanced subfamilies of the middle layer $\binom{[2n]}{n}$ of the Boolean lattice $2^{[2n]}$. A family $\mathcal{F}\subseteq\binom{[2n]}{n}$ is said to be balanced if every element in $[2n]$ appears in the same number of members of $\mathcal{F}$. A balanced subfamily of size 2 is exactly a complementary pair $\{A,[2n]\setminus A\}$, and therefore a family with no balanced subfamily of size $2$ has at most $\frac{1}{2}\binom{2n}{n}$ members. We show that for every $k\geq 1$ and all sufficiently large $n$, this maximum size is compatible with delaying the smallest size of a balanced subfamily until $2k+2$. More precisely, there exists a family $\mathcal{F}\subseteq\binom{[2n]}{n}$ of size $\frac{1}{2}\binom{2n}{n}$ with no balanced subfamilies of sizes $2,4,\ldots,2k$, but with a balanced subfamily of size $2k+2$. The proof is constructive and is obtained by lifting Taylor-Zwicker trade-robust magic-square games to self-dual selectors in the middle layer. This proves a recent conjecture of Moss and Pedersen.