English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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

On the maximum degree of induced subgraphs of the Kneser graph