Forbidding a Set Difference of Size 1

TL;DR

This paper establishes upper bounds on the size of set families avoiding pairs with a difference of exactly one element, showing these bounds are tight up to a constant factor, and extends results to differences of size k.

Contribution

It proves tight bounds on the maximum size of families avoiding specific set differences, generalizing to differences of size k.

Findings

Families avoiding |A extbackslash B|=1 are at most (2+o(1))/n times the middle binomial coefficient.

The bounds are tight up to a constant factor of 2.

Similar bounds are obtained for differences of size k, with size at most C_k/n^k times the middle binomial coefficient.

Abstract

How large can a family \cal A \subset \cal P [n] be if it does not contain A,B with |A\setminus B| = 1? Our aim in this paper is to show that any such family has size at most \frac{2+o(1)}{n} \binom {n}{\lfloor n/2\rfloor }. This is tight up to a multiplicative constant of $2$. We also obtain similar results for families \cal A \subset \cal P[n] with |A\setminus B| \neq k, showing that they satisfy |{\mathcal A}| \leq \frac{C_k}{n^k}\binom {n}{\lfloor n/2\rfloor }, where C_k is a constant depending only on k.