PreprintCC BY 4.0arXiv
The Three-Dimensional Erdős Box Problem Has Exponent 11/4
Abstract
Let z(n) be the maximum number of edges in a tripartite $3$-uniform hypergraph with n vertices in each part and no copy of K2,2,2(3) (a ``box''). Erdős (1964) proved that z(n) = O(n11/4), whereas the best previous lower bound, due to Katz, Krop, and Maggioni (2002), was Ω(n8/3). For each q = 2m, we construct a box-free hypergraph with q4 vertices in each part and q11 edges, showing that z(n) = Θ(n11/4). The construction uses the power map τ(s) = sq2−q+1 on Fq3, which sends the fibers of s ↦ τ(s+1) + τ(s) to pairwise skew affine lines over Fq.
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BibTeX
@misc{menezes2026threedimensional,
title = {{The Three-Dimensional Erdős Box Problem Has Exponent $11/4$}},
author = {Dean Menezes},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.22381},
eprint = {2609.22381},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22381},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Three-Dimensional Erdős Box Problem Has Exponent 11/4
AU - Dean Menezes
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2609.22381
UR - https://arxiv.org/abs/2609.22381
AB - Let z(n) be the maximum number of edges in a tripartite $3$-uniform hypergraph with n vertices in each part and no copy of K_2,2,2^(3) (a ``box''). Erdős (1964) proved that z(n) = O(n^11/4), whereas the best previous lower bound, due to Katz, Krop, and Maggioni (2002), was Ω(n^8/3). For each q = 2^m, we construct a box-free hypergraph with q^4 vertices in each part and q^11 edges, showing that z(n) = Θ(n^11/4). The construction uses the power map τ(s) = s^q^2−q+1 on F_q^3, which sends the fibers of s ↦ τ(s+1) + τ(s) to pairwise skew affine lines over F_q.
ER -