Set families with a forbidden pattern
Ilan Karpas, Eoin Long

TL;DR
This paper investigates the maximum size of set families avoiding certain balanced sign patterns in symmetric differences, establishing bounds that depend on the pattern's structure and size relative to n.
Contribution
It introduces new extremal bounds for P-free families in the Boolean lattice, especially for patterns with bounded balancedness and specific sign arrangements.
Findings
For fixed c>0, P-free families are o(2^n) when d < c log log n.
Stronger bounds are established for patterns with all pluses then minuses, and alternating signs, when d = o(√n).
Bounds are tight for patterns with all pluses followed by minuses.
Abstract
A balanced pattern of order is an element , where both signs appear times. Two sets form -pattern, which we denote by , if with and . We say is -free if for all . We consider the following extremal question: how large can a family be if is -free? We prove a number of results on the sizes of such families. In particular, we show that for some fixed , if is a -balanced pattern with then . We then give stronger bounds in the cases when (i) consists of signs, followed by signs…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsLimits and Structures in Graph Theory · Analytic Number Theory Research · graph theory and CDMA systems
