Projections of antichains (Q2309224)

Summary: A subset \(A\) of \(\mathbb{Z}^n\) is called a weak antichain if it does not contain two elements \(x\) and \(y\) satisfying \(x_i<y_i\) for all \(i\). \textit{K. Engel} et al. [``Projection inequalities for antichains'', Preprint, \url{arXiv:1812.06496}] showed that for any weak antichain \(A\), the sum of the sizes of its \((n-1)\)-dimensional projections must be at least as large as its size \(|A|\). They asked what the smallest possible value of the gap between these two quantities is in terms of \(|A|\). We answer this question by giving an explicit weak antichain attaining this minimum for each possible value of \(|A|\). In particular, we show that sets of the form \(A_N=\{x\in\mathbb{Z}^n: 0\leq x_j\leq N-1 \text{ for all \(j\) and } x_i=0\text{ for some \(i\)}\}\) minimise the gap among weak antichains of size \(|A_N|\).