P. Frankl

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20112026

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pairwise disjoint sets

15 citations · 86 across the 50 of their papers we have counts for

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61 papers · 1 filter

math.CO2026

Intersecting families with covering number three II

Peter Frankl, Jian Wang

A family $F \subset (k [ n ])$ is called intersecting if $F \cap F^{'} \neq = \emptyset$ for all $F, F^{'} \in F$ . The covering number of a family $F$ is defi…

math.CO2023

Odd-Sunflowers

Peter Frankl, János Pach, Dömötör Pálvölgyi

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 o…

math.CO2023

On the $C$ -diversity of intersecting hypergraphs

Peter Frankl, Jian Wang

Let $F \subset (k X)$ be a family consisting of $k$ -subsets of the $n$ -set $X$ . Suppose that $F$ is intersecting, i.e., $F \cap F^{'} \neq = \emptyset$ for all $…

math.CO2023

Variations on the Bollobás set-pair theorem

Gábor Hegedüs, Péter Frankl

Let $X$ be an $n$ -element set. A set-pair system $\mbox{$\cal P$}=\{(A_i,B_i)\}_{1\leq i\leq m}$ is a collection of pairs of disjoint subsets of $X$ . It is called skew Bollobás sys…

math.CO2023

Improved bounds on the maximum diversity of intersecting families

Peter Frankl, Jian Wang

A family $F \subset (k [ n ])$ is called an intersecting family if $F \cap F^{'} \neq = \emptyset$ for all $F, F^{'} \in F$ . If $\cap F \neq = \emptyset$ then $\…

math.CO2023

On asymptotic local Turán problems

Peter Frankl, Jiaxi Nie

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 $n$ -ve…

math.CO2023

Four-vertex traces of finite sets

Peter Frankl, Jian Wang

Let $[n] = X_{1} \cup X_{2} \cup X_{3}$ be a partition with $⌊ \frac{n}{3} ⌋ \leq ∣ X_{i} ∣ \leq ⌈ \frac{n}{3} ⌉$ and define $\mathcal{G}=\{G\subset [n]\colon |G\cap X_i|\leq 1,…

math.CO2022★ 3 cited

Best possible bounds on the double-diversity of intersecting hypergraphs

Peter Frankl, Jian Wang

For a family $F \subset (k [ n ])$ and two elements $x, y \in [n]$ define $F (x ˉ, y ˉ) = {F \in F : x \in / F, y \in / F}$ . The double-di…

math.CO2022★ 1 cited

Turán graphs with bounded matching number

Noga Alon, Peter Frankl

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 parameter…

math.CO2022

Improved bounds concerning the maximum degree of intersecting hypergraphs

Peter Frankl, Jian Wang

For positive integers $n > k > t$ let $(k [ n ])$ denote the collection of all $k$ -subsets of the standard $n$ -element set $[n] = {1, \dots, n}$ . Subsets of $(k [ n ])$ are ca…

math.CO2022

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

Peter Frankl

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.…

math.CO2022★ 1 cited

Intersecting families with covering number three

Peter Frankl, Jian Wang

We consider $k$ -graphs on $n$ vertices, that is, $F \subset (k [ n ])$ . A $k$ -graph $F$ is called intersecting if $F \cap F^{'} \neq = \emptyset$ for all $F,F'\in…

math.CO2022

A Product Version of the Hilton-Milner Theorem

Peter Frankl, Jian Wang

Two families $F, G$ of $k$ -subsets of ${1, 2, \dots, n}$ are called non-trivial cross-intersecting if $F \cap G \neq = \emptyset$ for all $F\in \mathcal{F}, G\in \ma…

math.CO2022

A Product Version of the Hilton-Milner-Frankl Theorem

Peter Frankl, Jian Wang

Two families $F, G$ of $k$ -subsets of ${1, 2, \dots, n}$ are called non-trivial cross $t$ -intersecting if $∣ F \cap G ∣ \geq t$ for all $F\in \mathcal{F}, G\in \math…

math.CO2022

Intersections and Distinct Intersections in Cross-intersecting Families

Peter Frankl, Jian Wang

Let $F, G$ be two cross-intersecting families of $k$ -subsets of ${1, 2, \dots, n}$ . Let $F \land G$ , $I (F, G)$ d…

math.CO2022★ 2 cited

Extremal results for graphs avoiding a rainbow subgraph

Peter Frankl, Ervin Győri, Zhen He +5

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 t…

math.CO2022★ 2 cited

Graphs without rainbow triangles

Peter Frankl

Let F,G,H be three graphs on the same n vertices. We consider the maximum of the sum and product of the number of their edges subject to the condition in the title.

math.CO2022

Perfect matchings in down-sets

Peter Frankl, Andrey Kupavskii

In this paper, we show that, given two down-sets (simplicial complexes) there is a matching between them that matches disjoint sets and covers the smaller of the two down-sets. Thi…

math.CO2021

Piercing the chessboard

Gergely Ambrus, Imre Bárány, Péter Frankl +1

We consider the minimum number of lines $h_{n}$ and $p_{n}$ needed to intersect or pierce, respectively, all the cells of the $n \times n$ chessboard. Determining these values can also…

math.CO2021

On the maximum number of distinct intersections in an intersecting family

Peter Frankl, Sergei Kiselev, Andrey Kupavskii

For $n > 2 k \geq 4$ we consider intersecting families $F$ consisting of $k$ -subsets of ${1, 2, \dots, n}$ . Let $I (F)$ denote the family of all distinc…

math.CO2021

Best possible bounds on the number of distinct differences in intersecting families

Peter Frankl, Sergei Kiselev, Andrey Kupavskii

For a family $F$ , let $D (F)$ stand for the family of all sets that can be expressed as $F ∖ G$ , where $F, G \in F$ . A family $F$ i…

math.CO2021

Uniform intersecting families with large covering number

Peter Frankl, Andrey Kupavskii

A family $F$ has covering number $τ$ if the size of the smallest set intersecting all sets from $F$ is equal to $τ$ . Let $M (n, k, τ)$ stand for the size of the larg…

math.CO2021

On the sum of sizes of overlapping families

Peter Frankl, Jian Wang

Let $A_{1}, \dots, A_{m}$ be families of $k$ -subsets of an $n$ -set. Suppose that one cannot choose pairwise disjoint edges from $s + 1$ distinct families. Subject to t…

math.CO2021

Exchange properties of finite set-systems

Peter Frankl, János Pach, Dömötör Pálvölgyi

In a recent breakthrough, Adiprasito, Avvakumov, and Karasev constructed a triangulation of the $n$ -dimensional real projective space with a subexponential number of vertices. They…

Intersecting families with covering number three II

Odd-Sunflowers

On the C-diversity of intersecting hypergraphs

Variations on the Bollobás set-pair theorem

Improved bounds on the maximum diversity of intersecting families

On asymptotic local Turán problems

Four-vertex traces of finite sets

Best possible bounds on the double-diversity of intersecting hypergraphs

Turán graphs with bounded matching number

Improved bounds concerning the maximum degree of intersecting hypergraphs

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

Intersecting families with covering number three

A Product Version of the Hilton-Milner Theorem

A Product Version of the Hilton-Milner-Frankl Theorem

Intersections and Distinct Intersections in Cross-intersecting Families

Extremal results for graphs avoiding a rainbow subgraph

Graphs without rainbow triangles

Perfect matchings in down-sets

Piercing the chessboard

On the maximum number of distinct intersections in an intersecting family

Best possible bounds on the number of distinct differences in intersecting families

Uniform intersecting families with large covering number

On the sum of sizes of overlapping families

Exchange properties of finite set-systems

On the $C$ -diversity of intersecting hypergraphs