A note on the size of $\mathscr {N}$-free families

Abstract

The $\mathscr {N}$ poset consists of four distinct sets W, X, Y, Z such that $W\subset X, Y\subset X$, and $Y\subset Z$ where W is not necessarily a subset of Z. A family ${{\mathscr {F}}}$, considered as a subposet of the n-dimensional Boolean lattice $\mathscr {B}_n$, is $\mathscr {N}$-free if it does not contain $\mathscr {N}$ as a subposet. Let $\mathrm{La}(n, \mathscr {N})$ be the size of a largest $\mathscr {N}$-free family in $\mathscr {B}_n$. Katona and Tarján proved that , where and is the size of a single-error-correcting code with constant weight $k+1$. In this note, we prove for n even and $k=n/2, \mathrm{La}(n, \mathscr {N}) \geqslant {n\atopwithdelims ()k}+A(n, 4, k)$, which improves the bound on $\mathrm{La}(n, \mathscr {N})$ in the second order term for some values of n and should be an improvement for an infinite family of values of n, depending on the behavior of the function .