An Approximate Vertex-Isoperimetric Inequality for rrr-sets (original) (raw)

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Keywords: discrete isoperimetric inequalities

Abstract

We prove a vertex-isoperimetric inequality for \([n]^{(r)}\), the set of all \(r\)-element subsets of \(\{1,2,\ldots,n\}\), where \(x,y \in [n]^{(r)}\) are adjacent if \(|x \Delta y|=2\). Namely, if \(\mathcal{A} \subset [n]^{(r)}\) with \(|\mathcal{A}|=\alpha {n \choose r}\), then its vertex-boundary \(b(\mathcal{A})\) satisfies
\[|b(\mathcal{A})| \geq c\sqrt{\frac{n}{r(n-r)}} \alpha(1-\alpha) {n \choose r},\]
where \(c\) is a positive absolute constant. For \(\alpha\) bounded away from 0 and 1, this is sharp up to a constant factor (independent of \(n\) and \(r\)).

How to Cite

Christofides, D., Ellis, D., & Keevash, P. (2013). An Approximate Vertex-Isoperimetric Inequality for rrr-sets. The Electronic Journal of Combinatorics, 20(4), #P15. https://doi.org/10.37236/2458