A note on the size of N-free families

Abstract

The $\mathcal{N}$ 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 $\mathcal{F}$ as a subposet of the $n$-dimensional Boolean lattice, ${\mathcal{B}}_n$, is $\mathcal{N}$-free if it does not contain $\mathcal{N}$ as a subposet. Let $\text{La}(n,\mathcal{N})$ be the size of a largest $\mathcal{N}$-free family in ${\mathcal{B}}_n$. Katona and Tarján proved that $\text{La}(n,\mathcal{N})\ge \left(\genfrac{}{}{0ex}{}{n}{k}\right)+A(n,4,k+1)$, where $k=\lfloor n/2\rfloor$ and $A(n,4,k+1)$ is the size of a single-error-correcting code with constant weight $k+1$. In this note, we prove for $n$ even and $k=n/2$, $\text{La}(n,\mathcal{N})\ge \left(\genfrac{}{}{0ex}{}{n}{k}\right)+A(n,4,k)$, which improves the bound on $\text{La}(n,\mathcal{N})$ 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 $A(n,4,\cdot )$.