We consider $k$-graphs, $\mathcal{F}\subset \binom{[n]}{k}$, $k\geq 3$. A $k$-graph is called intersecting if any two of its edges have non-empty intersection. It is called a star if all its edges share a common vertex. The $k$-graph $\mathcal{F}$ is called Helly if all its intersecting subfamilies are stars. If the same is required only for subfamilies consisting of three edges, it is called triangle-free. It is well known that for $n\geq 3k/2$, the full star is the unique largest triangle-free family whence the largest Helly family as well. In 1984 Tuza proved the best possible bound $|\mathcal{F}|\leq \binom{n-k-1}{k-1}+\binom{n-2}{k-2}+1$ for Helly families that are not stars, albeit only for some unspecified $n>n_0(k)$. The aim of this paper is twofold. First we establish the same bound for $n>2k$. Second we show that for $n>12k^2$ the same upper bound holds for triangle-free families. It is shown as well that it is not true for $2k
For a fixed integer $k\ge 2$, let $W_{2k+1}=K_1\vee C_{2k}$ be an odd wheel graph. The fixed-size spectral extremal problem aims to determine \[ \operatorname{spex}(m,W_{2k+1}):=\max\{\rho(G): e(G)=m,\ G \text{ is } W_{2k+1}\text{-free}\}, \] where $\rho(G)$ denotes the adjacency spectral radius. Based on this problem,...
Let $G$ be a graph together with a total order $\prec$ on its edges. We say that $\prec$ is realizable in $\mathbb{R}^d$ if there is a placement of the vertices of $G$ in $\mathbb{R}^d$ such that the Euclidean lengths of the edges induce exactly the order $\prec$. Almendra-Hern\'andez and Mart\'inez-Sandoval proved tha...
Gerardo L. Maldonado, Leonardo Martínez-Sandoval, Miguel Raggi et al.· 0 citations
Let $\mathrm{Sym(n,k)}$ denote the set of permutations on $\{1,2,\ldots,n\}$ with exactly $k$ cycles. A family $\mathcal{F}\subset\mathrm{Sym}(n,k)$ is said to be intersecting if $\sigma^{-1}\tau$ has a fixed point for all $\sigma,\tau\in\mathcal{F}$. In this paper, we investigate the size and structure of maximum-size...
A digraph is antidirected if every vertex has indegree zero or outdegree zero. Let $k\ge2$, and let $F$ be an antidirected forest with $k$ arcs and no isolated vertices. We prove that every digraph $D$ of order $n$ with more than $g_k(n):=2\max\left\{\binom{2k-1}{2}, (k-1)\left(n-\frac{k}{2}\right)\right\}$ arcs contai...
Let $k\ge 3$ and let $n=km$. A $K_k$-factor in an $n$-vertex graph is a collection of $m$ vertex-disjoint copies of $K_k$ that covers the entire vertex set. We determine the maximum adjacency spectral radius of an $n$-vertex graph containing no $K_k$-factor when $m\ge 2k-1$. More precisely, we prove that every such gra...
We prove an incidence bound for bipartite graphs on finite subsets of $\mathbb{F}^2\times \mathbb{F}^2$ defined by Boolean combinations of polynomial equations of bounded degree. If such a graph is $K_{k,k}$-free and its vertex classes have sizes $m$ and $n$, then it has $O_{t,k}((mn)^{2/3}+m+n+mn/p)$ edges, where $t$...
Le Quang-Ham· 0 citations
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