On $γ$-vectors satisfying the Kruskal-Katona inequalities

The Kruskal-Katona Theorem for Graphs

In graph theory, knowing the number of complete subgraphs with r vertices that a graph g has, limits the number of its complete subgraphs with s vertices, for s > r. A useful upper bound is provided by the Kruskal-Katona theorem, but this bound is often not tight. In this note, we add to the known cases where this bound is tight and also investigate cases where it is not. Finally we look at a useful technique for actually finding the numbers of complete subgraphs of a graph.

Kruskal--Katona type theorems for clique complexes arising from chordal and strongly chordal graphs

A forest is the clique complex of a strongly chordal graph and a quasi-forest is the clique complex of a chordal graph. Kruskal--Katona type theorems for forests, quasi-forests, pure forests and pure quasi-forests will be presented. In addition, it will be shown that a quasi-forest is shellable if and only if its $h$-vector $(h_0, h_1, h_2, ...)$ satisfies $h_i = 0$ for $i > 1$.

Kruskal-Katona type Problem

The Kruskal Katona theorem was proved in the 1960s. In the theorem, we are given an integer $r$ and families of sets $\mathcal{A}\subset \mathbb{N}^{(r)}$ and $\mathcal{B}\subset\mathbb{N}^{(r-1)}$ such that for every $A\in\mathcal{A}$, every subset of $A$ of size $r-1$ is in $\mathcal{B}$. We are interested in finding the mimimum size of $b=|\mathcal{B}|$ given fixed values of $r$ and $a=|\mathcal{A}|$. The Kruskal Katona theorem states that this mimimum occurs when both $\mathcal{A}$ and $\mathcal{B}$ are initial segments of the colexicographic ordering. The Kruskal Katona theorem is very us…

Kruskal-Katona's function and a variation of cross-intersecting antichains

We prove some properties of the Kruskal-Katona function, and apply to the following variation of cross-intersecting antichains. Let $n\ge 4$ be an even integer and $\mathscr{A}$ and $\mathscr{B}$ be two cross-intersecting antichains of $\mathbb{N}_n$ with at most $k$ disjoint pairs, i.e. for all $A_i\in \mathscr{A}$, $B_j\in\mathscr{B}$, $A_i\cap B_j=\emptyset$ only if $i=j\le k$. We prove a best possible upper bound on $|\mathscr{A}|+|\mathscr{B}|$. Furthermore, we show that the extremal families contain only $\frac{n}{2}$ and $(\frac{n}{2}+1)$-sets.

A Kruskal-Katona type result and applications

Inspired by the Kruskal-Katona theorem a minimization problem is studied, where the role of the shadow is replaced by the image of the action of the monoid of increasing functions. One of our main results shows that compressed sets are a solution to this problem. Several applications to simplicial complexes are discussed.

A grid generalisation of the Kruskal-Katona theorem

For a set $A\subseteq\left[k\right]^{n}=\left\{ 0,\dots,k-1\right\} ^{n}$, we define the $d$-shadow of $A$ to be the set of points obtained by flipping to zero one of the non-zero coordinates of some point in $A$. Let $\left[k\right]_{r}^{n}$ be the set of those points in $\left[k\right]^{n}$ with exactly $r$ non-zero coordinates. Given the size of $A$, how should we choose $A\subseteq\left[k\right]_{r}^{n}$ so as to minimise the $d$-shadow? Note that the case $k=2$ is answered by the Kruskal-Katona theorem. Our aim in this paper is to give an exact answer to this question. In particular, we s…

关于彩色Kruskal--Katona定理

On Colorful Kruskal--Katona Theorems

该研究利用熵技术确定了边染色图中彩虹三角形的最大数量上界，并揭示了最优构造与仿射平面的关联，同时探讨了更大团计数的奇偶性差异。

Kruskal–Katona 定理的极值族

Extremal families for the Kruskal--Katona theorem

本文通过证明一个新的组合不等式，给出 Kruskal–Katona 定理的简单证明，并刻画阴影最小化问题的极值族，回答了 Füredi 和 Griggs 的公开问题。

基于熵方法的Kruskal--Katona型问题

Kruskal--Katona-Type Problems via the Entropy Method

本文利用熵方法研究给定边数条件下固定子图最大副本数的Kruskal--Katona型极值问题，证明了3-边着色图中彩虹三角形数量至多为$\sqrt{2RGB}$的紧界，并给出蕴含多个先前结果的Kruskal--Katona定理推广。

3-均匀超图最小度克鲁斯卡尔-卡托纳问题中孤立团的二次顶点阈值

A Quadratic Vertex Threshold for Isolated Cliques in the Minimum Degree Kruskal-Katona Problem for 3-Uniform Hypergraphs

研究3-均匀超图最小度克鲁斯卡尔-卡托纳问题，通过图变换正则化邻域结构，将问题简化为计数论证，证明对足够大的\(t\)，当\(|X|\)满足一定条件时，极值超图含\(K_{t + 1}^3\)孤立副本，给出二次阶阈值。

Kruskal-Katona for convex sets, with applications

The well-known Kruskal-Katona theorem in combinatorics says that (under mild conditions) every monotone Boolean function $f: \{0,1\}^n \to \{0,1\}$ has a nontrivial "density increment." This means that the fraction of inputs of Hamming weight $k+1$ for which $f=1$ is significantly larger than the fraction of inputs of Hamming weight $k$ for which $f=1.$ We prove an analogous statement for convex sets. Informally, our main result says that (under mild conditions) every convex set $K \subset \mathbb{R}^n$ has a nontrivial density increment. This means that the fraction of the radius-$r$ sphere t…

关于 Kruskal--Katona 定理的极族

On the extremal families for the Kruskal--Katona theorem

本文进一步发展了Kruskal-Katona定理极族的性质，推广了Füredi-Griggs/Mörs定理，证明了极族数量随k-二项式分解系数间隙严格增长，并给出了快速判定算法及新的等式刻画。

面向任意高维扩展器的KKL定理

Toward a KKL Theorem for any HDX

本文提出一种局部到全局方法，证明任何链接满足KKL定理的单纯复形整体也满足KKL型定理，并给出组合高维扩展器上非扩展函数的首次刻画。