On the maximum degree of induced subgraphs of the Kneser graph

Abstract

For integers $n\ge k\ge 1$, the {\em Kneser graph} $K(n,k)$ is the graph with vertex-set consisting of all the $k$-element subsets of $\{1,2,\dots ,n\}$, where two $k$-element sets are adjacent in $K(n,k)$ if they are disjoint. We show that if $(n,k,s)\in {\mathbb{N}}^3$ with $n>10000k{s}^5$ and $\mathcal{F}$ is set of vertices of $K(n,k)$ of size larger than $\{A\subset \{1,2,\dots ,n\}:|A|=k,A\cap \{1,2,\dots ,s\}\ne \varnothing \}$, then the subgraph of $K(n,k)$ induced by $\mathcal{F}$ has maximum degree at least \[ \left(1 - O\left(\sqrt{s^3 k/n}\right)\right)\frac{s}{s+1} \cdot {n-k \choose k} \cdot \frac{|\mathcal{F}|}{\binom{n}{k}}.\] This is sharp up to the behaviour of the error term $O(\sqrt{s^{3}k/n})$. In particular, if the triple of integers $(n,k,s)$ satisfies the condition above, then the minimum maximum degree does not increase `continuously' with $|\mathcal{F}|$. Instead, it has $s$ jumps, one at each time when $|\mathcal{F}|$ becomes just larger than the union of $i$ stars, for $i=1,2,\dots ,s$. An appealing special case of the above result is that if $\mathcal{F}$ is a family of $k$-element subsets of $\{1,2,\dots ,n\}$ with $|\mathcal{F}|=\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)+1$, then there exists $A\in \mathcal{F}$ such that $\mathcal{F}$ is disjoint from at least $$\left(1/2-O\left(\sqrt{k/n}\right)\right)\left(\genfrac{}{}{0ex}{}{n-k-1}{k-1}\right)$$ of the other sets in $\mathcal{F}$; this is asymptotically sharp if $k=o(n)$. Frankl and Kupavskii, using different methods, have recently proven closely related results under the hypothesis that $n$ is at least quadratic in $k$.

31 pages. Clarifications in response to comments of two anonymous referees