A note on the size of $$\mathscr {N}$$ 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 Tarjan proved that Open image in new window, where Open image in new window and Open image in new window 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 Open image in new window.