arXiv subjects

Peter Frankl

Publications and source records attributed to Peter Frankl.

At least 37 records · Page 2Linked to original sources

arXiv2024

On $r$-wise $t$-intersecting uniform families

We consider families, $\mathcal{F}$ of $k$-subsets of an $n$-set. For integers $r\geq 2$, $t\geq 1$, $\mathcal{F}$ is called $r$-wise $t$-intersecting if any $r$ of its members have at least $t$ elements in common. The most natural construction of such a family is the full $t$-star, consisting of all $k$-sets containing a fixed $t$-set. In the case $r=2$ the Exact Erdős-Ko-Rado Theorem shows that the full $t$-star is largest if $n\geq (t+1)(k-t+1)$. In the present paper, we prove that for $n\geq (2.5t)^{1/(r-1)}(k-t)+k$, the full $t$-star is largest in case of $r\geq 3$. Examples show that the exponent $\frac{1}{r-1}$ is best possible. This represents a considerable improvement on a recent result of Balogh and Linz.

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arXiv2024

Intersection problems and a correlation inequality for integer sequences

Let us consider a collection $\mathcal G$ of codewords of length $n$ over an alphabet of size $s$. Let $t_1,\ldots, t_s$ be nonnegative integers. What is the maximum of $|\mathcal G|$ subject to the condition that any two codewords should have at least $t_i$ positions where both have letter $i$ ($1\le i\le s$). In the case $s=2$ it is a longstanding open question. Quite surprisingly, we obtain an almost complete answer for $s\ge 3$. The main tool is a correlation inequality.

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arXiv2024

Intersecting families with large shadow degree

A $k$-uniform family $\mathcal{F}$ is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. The shadow family $\partial \mathcal{F}$ is the family of $(k-1)$-element sets that are contained in some members of $\mathcal{F}$. The shadow degree (or minimum positive co-degree) of $\mathcal{F}$ is defined as the maximum integer $r$ such that every $E\in \partial \mathcal{F}$ is contained in at least $r$ members of $\mathcal{F}$. In 2021, Balogh, Lemons and Palmer determined the maximum size of an intersecting $k$-uniform family with shadow degree at least $r$ for $n\geq n_0(k,r)$, where $n_0(k,r)$ is doubly exponential in $k$ for $4\leq r\leq k$. In the present paper, we present a short proof of this result for $n\geq 2(r+1)^rk \frac{\binom{2k-1}{k}}{\binom{2r-1}{r}}$ and $4\leq r\leq k$.

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arXiv2024

Triangle-free triple systems

There are four non-isomorphic configurations of triples that can form a triangle in a $3$-uniform hypergraph. Forbidding different combinations of these four configurations, fifteen extremal problems can be defined, several of which already appeared in the literature in some different context. Here we systematically study all of these problems solving the new cases exactly or asymptotically. In many cases we also characterize the extremal constructions.

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arXiv2024

Odd-Sunflowers

Extending the notion of sunflowers, we call a family of at least two sets an odd-sunflower if every element of the underlying set is contained in an odd number of sets or in none of them. It follows from the Erd\H os--Szemerédi conjecture, recently proved by Naslund and Sawin, that there is a constant $μ<2$ such that every family of subsets of an $n$-element set that contains no odd-sunflower consists of at most $μ^n$ sets. We construct such families of size at least $1.5021^n$. We also characterize minimal odd-sunflowers of triples.

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arXiv2023

On the Holroyd-Talbot Conjecture for Sparse Graphs

Given a graph $G$, let $μ(G)$ denote the size of the smallest maximal independent set in $G$. A family of subsets is called a star if some element is in every set of the family. A split vertex has degree at least 3. Holroyd and Talbot conjectured the following Erdős-Ko-Rado type statement about intersecting families of independent sets in graphs: if $1\le r\le μ(G)/2$ then there is an intersecting family of independent $r$-sets of maximum size that is a star. In this paper we prove similar statements for sparse graphs on $n$ vertices: roughly, for graphs of bounded average degree with $r\le O(n^{1/3})$, for graphs of bounded degree with $r\le O(n^{1/2})$, and for trees having a bounded number of split vertices with $r\le O(n^{1/2})$.

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arXiv2023

On asymptotic local Turán problems

An $r$-uniform hypergraph has $(q,p)$-property if any set of $q$ vertices spans a complete sub-hypergraph on $p$ vertices. Let $t_r(n,q,p)$ be the minimum edge density of an $n$-vertex $r$-uniform hypergraph with {\em $(q,p)$-property} and let $t_r(q,p)=\lim_{n\to\infty}t_r(n,q,p)$. A disjoint union of $k$ complete hypergraphs has $(q,\lceil q/k\rceil)$-property, which gives $t_r((q,\lceil{q/k}\rceil))\le 1/k^{r-1}$. The first author, Huang and Rödl showed that these constructions are the best asymptotically, that is, $\lim_{q\to\infty}t_r((q,\lceil{q/k}\rceil))=1/k^{r-1}$. They asked whether it is true for all real number $γ\ge1$ that $\lim_{q\to\infty}t_r((q,\lceil{q/γ}\rceil))=1/\lfloorγ\rfloor^{r-1}$. In this paper, we give positive answers to this question for a small range of real numbers, and, on the other hand, provide new constructions that give negative answers for many other ranges.

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arXiv2023

Uniform intersecting families with large covering number

A family $\mathcal F$ has covering number $τ$ if the size of the smallest set intersecting all sets from $\mathcal F$ is equal to $τ$. Let $M(n,k,τ)$ stand for the size of the largest intersecting family $\mathcal F$ of $k$-element subsets of $\{1,\ldots,n\}$ with covering number $τ$. It is a classical result of Erd\H os and Lovász that $M(n,k,k)\le k^k$ for any $n$. In this short note, we explore the behaviour of $M(n,k,τ)$ for $n k-\frac 12k^{1/2}$.

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arXiv2023

Improved bounds on the maximum diversity of intersecting families

A family $\mathcal{F}\subset \binom{[n]}{k}$ is called an intersecting family if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. If $\cap \mathcal{F}\neq \emptyset$ then $\mathcal{F}$ is called a star. The diversity of an intersecting family $\mathcal{F}$ is defined as the minimum number of $k$-sets in $\mathcal{F}$, whose deletion results in a star. In the present paper, we prove that for $n>36k$ any intersecting family $\mathcal{F}\subset \binom{[n]}{k}$ has diversity at most $\binom{n-3}{k-2}$, which improves the previous best bound $n>72k$ due to the first author. This result is derived from some strong bounds concerning the maximum degree of large intersecting families. Some related results are established as well.

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arXiv2023

Four-vertex traces of finite sets

Let $[n]=X_1\cup X_2\cup X_3$ be a partition with $\lfloor\frac{n}{3}\rfloor \leq |X_i|\leq \lceil\frac{n}{3}\rceil$ and define $\mathcal{G}=\{G\subset [n]\colon |G\cap X_i|\leq 1, 1\leq i\leq 3\}$. It is easy to check that the trace $\mathcal{G}_{\mid Y}:=\{G\cap Y\colon G\in \mathcal{G}\}$ satisfies $|\mathcal{G}_{\mid Y}|\leq 12$ for all 4-sets $Y\subset [n]$. For $n\geq 25$ it is proven that whenever $\mathcal{F}\subset 2^{[n]}$ satisfies $|\mathcal{F}|>|\mathcal{G}|$ then $|\mathcal{F}_{\mid C}|\geq 13$ for some $C\subset [n]$, $|C|=4$. Several further results of a similar flavor are established as well.

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arXiv2022

Best possible bounds on the double-diversity of intersecting hypergraphs

For a family $\mathcal{F}\subset \binom{[n]}{k}$ and two elements $x,y\in [n]$ define $\mathcal{F}(\bar{x},\bar{y})=\{F\in \mathcal{F}\colon x\notin F,\ y\notin F\}$. The double-diversity $γ_2(\mathcal{F})$ is defined as the minimum of $|\mathcal{F}(\bar{x},\bar{y})|$ over all pairs $x,y$. Let $\mathcal{L}\subset\binom{[7]}{3}$ consist of the seven lines of the Fano plane. For $n\geq 7$, $k\geq 3$ one defines the Fano $k$-graph $\mathcal{F}_{\mathcal{L}}$ as the collection of all $k$-subsets of $[n]$ that contain at least one line. It is proven that for $n\geq 13k^2$ the Fano $k$-graph is the essentially unique family maximizing the double diversity over all $k$-graphs without a pair of disjoint edges. Some similar, although less exact results are proven for triple and higher diversity as well.

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arXiv2022

Cells in the box and a hyperplane

It is well-known that a line can intersect at most $2n-1$ cells of the $n \times n$ chessboard. Here we consider the high dimensional version: how many cells of the $d$-dimensional $n\times \ldots \times n$ box can a hyperplane intersect? We also prove the lattice analogue of the following well-known fact. If $K,L$ are convex bodies in $R^d$ and $K\subset L$, then the surface area of $K$ is smaller than that of $L$.

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arXiv2022

A Product Version of the Hilton-Milner-Frankl Theorem

Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\ldots,n\}$ are called non-trivial cross $t$-intersecting if $|F\cap G|\geq t$ for all $F\in \mathcal{F}, G\in \mathcal{G}$ and $|\cap \{F\colon F\in \mathcal{F}\}|<t$, $|\cap \{G\colon G\in\mathcal{G}\}|<t$. In the present paper, we determine the maximum product of the sizes of two non-trivial cross $t$-intersecting families of $k$-subsets of $\{1,2,\ldots,n\}$ for $n\geq 4(t+2)^2k^2$, $k\geq 5$, which is a product version of the Hilton-Milner-Frankl Theorem.

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arXiv2022

Extremal results for graphs avoiding a rainbow subgraph

We say that $k$ graphs $G_1,G_2,\dots,G_k$ on a common vertex set of size $n$ contain a rainbow copy of a graph $H$ if their union contains a copy of $H$ with each edge belonging to a distinct $G_i$. We provide a counterexample to a conjecture of Frankl on the maximum product of the sizes of the edge sets of three graphs avoiding a rainbow triangle. We propose an alternative conjecture, which we prove under the additional assumption that the union of the three graphs is complete. Furthermore, we determine the maximum product of the sizes of the edge sets of three graphs or four graphs avoiding a rainbow path of length three.

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arXiv2022

Turán graphs with bounded matching number

We determine the maximum possible number of edges of a graph with $n$ vertices, matching number at most $s$ and clique number at most $k$ for all admissible values of the parameters.

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arXiv2022

Improved bounds concerning the maximum degree of intersecting hypergraphs

For positive integers $n>k>t$ let $\binom{[n]}{k}$ denote the collection of all $k$-subsets of the standard $n$-element set $[n]=\{1,\ldots,n\}$. Subsets of $\binom{[n]}{k}$ are called $k$-graphs. A $k$-graph $\mathcal{F}$ is called $t$-intersecting if $|F\cap F'|\geq t$ for all $F,F'\in \mathcal{F}$. One of the central results of extremal set theory is the Erdős-Ko-Rado Theorem which states that for $n\geq (k-t+1)(t+1)$ no $t$-intersecting $k$-graph has more than $\binom{n-t}{k-t}$ edges. For $n$ greater than this threshold the $t$-star (all $k$-sets containing a fixed $t$-set) is the only family attaining this bound. Define $\mathcal{F}(i)=\{F\setminus \{i\}\colon i\in F\in \mathcal{F}\}$. The quantity $\varrho(\mathcal{F})=\max\limits_{1\leq i\leq n}|\mathcal{F}(i)|/|\mathcal{F}|$ measures how close a $k$-graph is to a star. The main result (Theorem 1.5) shows that $\varrho(\mathcal{F})>1/d$ holds if $\mathcal{F}$ is 1-intersecting, $|\mathcal{F}|>2^dd^{2d+1}\binom{n-d-1}{k-d-1}$ and $n\geq 4(d-1)dk$. Such a statement can be deduced from the results of \cite{F78-2} and \cite{DF}, however only for much larger values of $n/k$ and/or $n$. The proof is purely combinatorial, it is based on a new method: shifting ad extremis. The same method is applied to obtain some nearly optimal bounds in the case of $t\geq 2$ (Theorem 1.11) along with a number of related results.

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arXiv2022

On the maximum of the sum of the sizes of non-trivial cross-intersecting families

We consider families of k-subsets of the standard n-set. Two families F, G are said to be cross-intersecting if every member of F has non-empty intersection with every member of G. A family is called non-trivial if the intersection of all its members is empty. Supposing that F and G are non-trivial and cross-intersecting, we determine the maximum of |F|+|G|. For the proof a strengthened version of the so-called shifting technique is introduced. The corresponding problem for families of different uniformities is solved as well.

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arXiv2022

A Product Version of the Hilton-Milner Theorem

Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\ldots,n\}$ are called non-trivial cross-intersecting if $F\cap G\neq \emptyset$ for all $F\in \mathcal{F}, G\in \mathcal{G}$ and $\cap \{F\colon F\in \mathcal{F}\}=\emptyset=\cap \{G\colon G\in \mathcal{G}\}$. In the present paper, we determine the maximum product of the sizes of two non-trivial cross-intersecting families of $k$-subsets of $\{1,2,\ldots,n\}$ for $n\geq 4k$, $k\geq 8$, which is a product version of the classical Hilton-Milner Theorem.

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