On balanced subfamilies of maximum complement-free families in the middle layer of the Boolean lattice
Abstract
We study balanced subfamilies of the middle layer binom[2n]n of the Boolean lattice 2[2n]. A family ℱ⊆binom[2n]n is said to be balanced if every element in [2n] appears in the same number of members of ℱ. A balanced subfamily of size 2 is exactly a complementary pair {A,[2n]∖ A}, and therefore a family with no balanced subfamily of size $2$ has at most 1/2binom2nn members. We show that for every k≥ 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 ℱ⊆binom[2n]n of size 1/2binom2nn with no balanced subfamilies of sizes 2,4,…,2k, but with a balanced subfamily of size 2k+2. The proof is constructive and uses a general lifting from simple games to self-dual selectors in the middle layer. This provides an efficient constructive proof of a recent conjecture of Moss and Pedersen.
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BibTeX
@misc{blanco2026balanced,
title = {{On balanced subfamilies of maximum complement-free families in the middle layer of the Boolean lattice}},
author = {Sa'ul A. Blanco},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.16172},
eprint = {2606.16172},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.16172},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On balanced subfamilies of maximum complement-free families in the middle layer of the Boolean lattice
AU - Sa'ul A. Blanco
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2606.16172
UR - https://arxiv.org/abs/2606.16172
AB - We study balanced subfamilies of the middle layer binom[2n]n of the Boolean lattice 2^[2n]. A family ℱ⊆binom[2n]n is said to be balanced if every element in [2n] appears in the same number of members of ℱ. A balanced subfamily of size 2 is exactly a complementary pair {A,[2n]∖ A}, and therefore a family with no balanced subfamily of size $2$ has at most 1/2binom2nn members. We show that for every k≥ 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 ℱ⊆binom[2n]n of size 1/2binom2nn with no balanced subfamilies of sizes 2,4,…,2k, but with a balanced subfamily of size 2k+2. The proof is constructive and uses a general lifting from simple games to self-dual selectors in the middle layer. This provides an efficient constructive proof of a recent conjecture of Moss and Pedersen.
ER -