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Peter Frankl

Publications and source records attributed to Peter Frankl.

At least 73 records · Page 4Linked to original sources
arXiv2020

Shattered matchings in intersecting hypergraphs

Let $X$ be an $n$-element set, where $n$ is even. We refute a conjecture of J. Gordon and Y. Teplitskaya, according to which, for every maximal intersecting family $\mathcal{F}$ of $\frac{n}2$-element subsets of $X$, one can partition $X$ into $\frac{n}2$ disjoint pairs in such a way that no matter how we pick one element from each of the first $\frac{n}2 - 1$ pairs, the set formed by them can always be completed to a member of $\mathcal{F}$ by adding an element of the last pair. The above problem is related to classical questions in extremal set theory. For any $t\ge 2$, we call a family of sets $\mathcal{F}\subset 2^X$ {\em $t$-separable} if for any ordered pair of elements $(x,y)$ of $X$, there exists $F\in\mathcal{F}$ such that $F\cap\{x,y\}=\{x\}$. For a fixed $t, 2\le t\le 5$ and $n\rightarrow\infty$, we establish asymptotically tight estimates for the smallest integer $s=s(n,t)$ such that every family $\mathcal{F}$ with $|\mathcal{F}|\ge s$ is $t$-separable.

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arXiv2020

On local Turán problems

Since its formulation, Turán's hypergraph problems have been among the most challenging open problems in extremal combinatorics. One of them is the following: given a $3$-uniform hypergraph $\mathcal{F}$ on $n$ vertices in which any five vertices span at least one edge, prove that $|\mathcal{F}| \ge (1/4 -o(1))\binom{n}{3}$. The construction showing that this bound would be best possible is simply $\binom{X}{3} \cup \binom{Y}{3}$ where $X$ and $Y$ evenly partition the vertex set. This construction has the following more general $(2p+1, p+1)$-property: any set of $2p+1$ vertices spans a complete sub-hypergraph on $p+1$ vertices. One of our main results says that, quite surprisingly, for all $p>2$ the $(2p+1,p+1)$-property implies the conjectured lower bound.

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arXiv2020

Maximal degrees in subgraphs of Kneser graphs

In this paper, we study the maximum degree in non-empty induced subgraphs of the Kneser graph $KG(n,k)$. One of the main results asserts that, for $k>k_0$ and $n>64k^2$, whenever a non-empty subgraph has $m\ge k{n-2\choose k-2}$ vertices, its maximum degree is at least $\frac 12(1-\frac {k^2}n) m - {n-2\choose k-2}\ge 0.49 m$. This bound is essentially best possible. One of the intermediate steps is to obtain structural results on non-empty subgraphs with small maximum degree.

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arXiv2020

Intersection theorems for $(-1,0,1)$-vectors

In this paper, we investigate Erd\H os--Ko--Rado type theorems for families of vectors from $\{0,\pm 1\}^n$ with fixed numbers of $+1$'s and $-1$'s. Scalar product plays the role of intersection size. In particular, we sharpen our earlier result on the largest size of a family of such vectors that avoids the smallest possible scalar product. We also obtain an exact result for the largest size of a family with no negative scalar products.

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arXiv2019

Intersection theorems for $\{0,\pm 1\}$-vectors and $s$-cross-intersecting families

In this paper we study two directions of extending the classical Erd\H os-Ko-Rado theorem which states that any family of $k$-element subsets of the set $[n] = \{1,\ldots,n\}$ in which any two sets intersect, has cardinality at most ${n-1\choose k-1}$. In the first part of the paper we study the families of $\{0,\pm 1\}$-vectors. Denote by $\mathcal L_k$ the family of all vectors $\mathbf v$ from $\{0,\pm 1\}^n$ such that $\langle\mathbf v,\mathbf v\rangle = k$. For any $k$, most $l$ and sufficiently large $n$ we determine the maximal size of the family $\mathcal V\subset \mathcal L_k$ such that for any $\mathbf v,\mathbf w\in \mathcal V$ we have $\langle \mathbf v,\mathbf w\rangle\ge l$. We find some exact values of this function for all $n$ for small values of $k$. In the second part of the paper we study cross-intersecting pairs of families. We say that two families are $\mathcal A, \mathcal B$ are \textit{$s$-cross-intersecting}, if for any $A\in\mathcal A,B\in \mathcal B$ we have $|A\cap B|\ge s$. We also say that a set family $\mathcal A$ is {\it $t$-intersecting}, if for any $A_1,A_2\in \mathcal A$ we have $|A_1\cap A_2|\ge t$. For a pair of nonempty $s$-cross-intersecting $t$-intersecting families $\mathcal A,\mathcal B$ of $k$-sets, we determine the maximal value of $|\mathcal A|+|\mathcal B|$ for $n$ sufficiently large.

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arXiv2019

Sharp results concerning disjoint cross-intersecting families

For an $n$-element set $X$ let $\binom{X}{k}$ be the collection of all its $k$-subsets. Two families of sets $\mathcal A$ and $\mathcal B$ are called cross-intersecting if $A\cap B \neq \emptyset$ holds for all $A\in\mathcal A$, $B\in\mathcal B$. Let $f(n,k)$ denote the maximum of $\min\{|\mathcal A|, |\mathcal B|\}$ where the maximum is taken over all pairs of {\em disjoint}, cross-intersecting families $\mathcal A, \mathcal B\subset\binom{[n]}{k}$. Let $c=\log_2e$. We prove that $f(n,k)=\left\lfloor\frac12\binom{n-1}{k-1}\right\rfloor$ essentially iff $n>ck^2$ (cf. Theorem~1.4 for the exact statement). Let $f^*(n,k)$ denote the same maximum under the additional restriction that the intersection of all members of both $\mathcal A$ and $\mathcal B$ are empty. For $k\ge5$ and $n\ge k^3$ we show that $f^*(n,k)=\left\lfloor\frac12\left(\binom{n-1}{k-1}-\binom{n-2k}{k-1}\right)\right\rfloor+1$ and the restriction on $n$ is essentially sharp (cf. Theorem~5.4).

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arXiv2019

Extremal set theory for the binomial norm

Best possible bounds are established for families without s pairwise disjoint members and the more general problem for several families. The results are shown to apply several classical results.

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arXiv2018

Diversity

Given a family $\mathcal F\subset 2^{[n]}$, its diversity is the number of sets not containing an element with the highest degree. The concept of diversity has proven to be very useful in the context of $k$-uniform intersecting families. In this paper, we study (different notions of) diversity in the context of other extremal set theory problems. One of the main results of the paper is a sharp stability result for cross-intersecting families in terms of diversity and, slightly more generally, sharp stability for the Kruskal--Katona theorem.

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arXiv2018

Two problems on matchings in set families - in the footsteps of Erdős and Kleitman

The families $\mathcal F_1,\ldots, \mathcal F_s\subset 2^{[n]}$ are called $q$-dependent if there are no pairwise disjoint $F_1\in \mathcal F_1,\ldots, F_s\in\mathcal F_s$ satisfying $|F_1\cup\ldots\cup F_s|\le q.$ We determine $\max |\mathcal F_1|+\ldots +|\mathcal F_s| $ for all values $n\ge q,s\ge 2$. The result provides a far-reaching generalization of an important classical result of Kleitman. The well-known Erd\H os Matching Conjecture suggests the largest size of a family $\mathcal F\subset {[n]\choose k}$ with no $s$ pairwise disjoint sets. After more than 50 years its full solution is still not in sight. In the present paper, we provide a Hilton-Milner-type stability theorem for the Erdős Matching Conjecture in a relatively wide range, in particular, for $n\ge (2+o(1))sk$ with $o(1)$ depending on $s$ only. This is a considerable improvement of a classical result due to Bollobás, Daykin and Erdős. We apply our results to advance in the following anti-Ramsey-type problem, proposed by Özkahya and Young. Let $ar(n,k,s)$ be the minimum number $x$ of colors such that in any coloring of the $k$-element subsets of $[n]$ with $x$ (non-empty) colors there is a \textit{rainbow matching} of size $s$, that is, $s$ sets of different colors that are pairwise disjoint. We prove a stability result for the problem, which allows to determine $ar(n,k,s)$ for all $k\ge 3$ and $n\ge sk+(s-1)(k-1).$ Some other consequences of our results are presented as well.

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arXiv2018

Families of sets with no matchings of sizes 3 and 4

In this paper, we study the following classical question of extremal set theory: what is the maximum size of a family of subsets of $[n]$ such that no $s$ sets from the family are pairwise disjoint? This problem was first posed by Erd\H os and resolved for $n\equiv 0, -1\ (\mathrm{mod }\ s)$ by Kleitman in the 60s. Very little progress was made on the problem until recently. The only result was a very lengthy resolution of the case $s=3,\ n\equiv 1\ (\mathrm{mod }\ 3)$ by Quinn, which was written in his PhD thesis and never published in a refereed journal. In this paper, we give another, much shorter proof of Quinn's result, as well as resolve the case $s=4,\ n\equiv 2\ (\mathrm{mod }\ 4)$. This complements the results in our recent paper, where, in particular, we answered the question in the case $n\equiv -2\ (\mathrm{mod }\ s)$ for $s\ge 5$.

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arXiv2018

Incompatible intersection properties

Let $\mathcal F\subset 2^{[n]}$ be a family in which any three sets have non-empty intersection and any two sets have at least $38$ elements in common. The nearly best possible bound $|\mathcal F|\le 2^{n-2}$ is proved. We believe that $38$ can be replaced by $3$ and provide a simple-looking conjecture that would imply this.

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arXiv2018

A stability theorem on cube tessellations

It is shown that if a $d$-dimensional cube is decomposed into n cubes, the side lengths of which belong to the interval $\left(1-\frac{1}{n^{1/d}+1}, 1\right], then $n$ is a perfect $d$-th power and all cubes are of the same size. This result is essentially tight.

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arXiv2017

Erd\H os-Ko-Rado theorem for $\{0,\pm 1\}$-vectors

The main object of this paper is to determine the maximum number of $\{0,\pm 1\}$-vectors subject to the following condition. All vectors have length $n$, exactly $k$ of the coordinates are $+1$ and one is $-1$, $n \geq 2k$. Moreover, there are no two vectors whose scalar product equals the possible minimum, $-2$. Thus, this problem may be seen as an extension of the classical Erd\H os-Ko-Rado theorem. Rather surprisingly there is a phase transition in the behaviour of the maximum at $n=k^2$. Nevertheless, our solution is complete. The main tools are from extremal set theory and some of them might be of independent interest.

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arXiv2017

Counting intersecting and pairs of cross-intersecting families

A family of subsets of $\{1,\ldots,n\}$ is called {\it intersecting} if any two of its sets intersect. A classical result in extremal combinatorics due to Erdős, Ko, and Rado determines the maximum size of an intersecting family of $k$-subsets of $\{1,\ldots, n\}$. In this paper we study the following problem: how many intersecting families of $k$-subsets of $\{1,\ldots, n\}$ are there? Improving a result of Balogh, Das, Delcourt, Liu, and Sharifzadeh, we determine this quantity asymptotically for $n\ge 2k+2+2\sqrt{k\log k}$ and $k\to \infty$. Moreover, under the same assumptions we also determine asymptotically the number of {\it non-trivial} intersecting families, that is, intersecting families for which the intersection of all sets is empty. We obtain analogous results for pairs of cross-intersecting families.

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arXiv2017

Families with no $s$ pairwise disjoint sets

For integers $n\ge s\ge 2$ let $e(n,s)$ denote the maximum of $|\mathcal F|,$ where $\mathcal F$ is a family of subsets of an $n$-element set and $\mathcal F$ contains no $s$ pairwise disjoint members. Half a century ago, solving a conjecture of Erd\H os, Kleitman determined $e(sm-1,s)$ and $e(sm,s)$ for all $m,s\ge 1$. During the years very little progress in the general case was made. In the present paper we state a general conjecture concerning the value of $e(sm-l,m)$ for $1 s_0(l,m).$ For $l=2$ we determine the value of $e(sm-2,m)$ for all $s\ge 5.$ Some related results shedding light on the problem from a more general context are proved as well.

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arXiv2017

Borsuk and Ramsey type questions in Euclidean space

We give a short survey of problems and results on (1) diameter graphs and hypergraphs, and (2) geometric Ramsey theory. We also make some modest contributions to both areas. Extending a well known theorem of Kahn and Kalai which disproved Borsuk's conjecture, we show that for any integer $r\ge 2$, there exist $\varepsilon=\varepsilon(r)>0$ and $d_0=d_0(r)$ with the following property. For every $d\ge d_0$, there is a finite point set $P\subset\mathbb{R}^d$ of diameter $1$ such that no matter how we color the elements of $P$ with fewer than $(1+\varepsilon)^{\sqrt{d}}$ colors, we can always find $r$ points of the same color, any two of which are at distance $1$.

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