Füredi's Conjecture and a Sharp Strengthening

Abstract

We prove a strengthening of Füredi's conjecture on strong Bollobás t-systems. For nonnegative integers t and s, a family of m≥2 pairs of finite sets (A_i,B_i) satisfying |A_i∩ B_i|≤ t and |A_i∩ B_j|>t+s for all i≠ j satisfies ∑_i=1^m (binom|B_i|−t+ss)/(binom|A_i|+|B_i|−2t|A_i|−t)≤1. When s=0, this is precisely Füredi's conjectured weighted inequality. The proof is based on a local growth inequality for graded exterior ideals, a basis-exchange construction, and a common-section reduction. The same method gives the corresponding inequality for subspaces over arbitrary fields. We also determine a best possible upper bound for the Füredi weight sum, construct equality examples with an arbitrary prescribed number of pairs, and derive further inequalities from two-sided basis exchanges.

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BibTeX RIS

BibTeX

@misc{huang2026füredis,
  title = {{Füredi's Conjecture and a Sharp Strengthening}},
  author = {Zihao Huang and Suijie Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11742},
  eprint = {2609.11742},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11742},
}

RIS

TY  - UNPB
TI  - Füredi's Conjecture and a Sharp Strengthening
AU  - Zihao Huang
AU  - Suijie Wang
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.11742
UR  - https://arxiv.org/abs/2609.11742
AB  - We prove a strengthening of Füredi's conjecture on strong Bollobás t-systems. For nonnegative integers t and s, a family of m≥2 pairs of finite sets (A_i,B_i) satisfying |A_i∩ B_i|≤ t and |A_i∩ B_j|>t+s for all i≠ j satisfies ∑_i=1^m (binom|B_i|−t+ss)/(binom|A_i|+|B_i|−2t|A_i|−t)≤1. When s=0, this is precisely Füredi's conjectured weighted inequality. The proof is based on a local growth inequality for graded exterior ideals, a basis-exchange construction, and a common-section reduction. The same method gives the corresponding inequality for subspaces over arbitrary fields. We also determine a best possible upper bound for the Füredi weight sum, construct equality examples with an arbitrary prescribed number of pairs, and derive further inequalities from two-sided basis exchanges.
ER  -

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