On the maximum degree of induced subgraphs of the Kneser graph
Hou Tin Chau, David Ellis, Ehud Friedgut, Noam Lifshitz
TL;DR
The paper investigates the maximum degree in subgraphs of the Kneser graph $K(n,k)$ induced by families $\mathcal{F} \subset { [n] \choose k}$, proving a sharp jump phenomenon: if $n > 10000 k s^5$ and $|\mathcal{F}|$ is not contained in a union of $s$ stars, the induced maximum degree is bounded below by a near-optimal expression depending on $|\mathcal{F}|$, with an error term $O\left(\sqrt{s^3 k/n}\right)$. The authors blend spectral methods with stability arguments, decomposing indicator functions along Kneser graph eigenspaces and applying the Expander Mixing Lemma to force the density pattern of $\mathcal{F}$ to align with unions of stars or to yield large edge counts otherwise. They present both a random construction and an explicit construction achieving small maximum degree for fixed size parameter $\lambda \in [s,s+1]$, and discuss a dense minimizer conjecture alongside a sparse minimizer framework. A key corollary mirrors Huang’s jump phenomenon in the hypercube, and the results tie into the Erdős matching conjecture by showing that, under the stated regime, any sufficiently large $\mathcal{F}$ must contain a matching of size $s+1$, connecting maximum-degree considerations to extremal matching behavior in the Kneser setting. The work thus provides sharp, quantifiable structure for induced subgraphs of $K(n,k)$ and bridges spectral techniques with combinatorial stability in a central extremal graph problem.
Abstract
For integers $n \geq k \geq 1$, the {\em Kneser graph} $K(n, k)$ is the graph with 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 they are 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 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 $i = 1, 2, \ldots, s$. An appealing special case of the above 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 there exists $A \in \mathcal{F}$ such that $\mathcal{F}$ is disjoint from at least $$\left(1/2-O\left(\sqrt{k/n}\right)\right){n-k-1 \choose k-1}$$ 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 $k$.