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Peter Frankl

Publications and source records attributed to Peter Frankl.

83 records · Page 5Linked to original sources
arXiv2017

Partition-free families of sets

Let $m(n)$ denote the maximum size of a family of subsets which does not contain two disjoint sets along with their union. In 1968 Kleitman proved that $m(n) = {n\choose m+1}+\ldots +{n\choose 2m+1}$ if $n=3m+1$. Confirming the conjecture of Kleitman, we establish the same equality for the cases $n=3m$ and $n=3m+2$, and also determine all extremal families. Unlike the case $n=3m+1$, the extremal families are not unique. This is a plausible reason behind the relative difficulty of our proofs. We completely settle the case of several families as well.

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arXiv2017

A degree version of the Hilton--Milner theorem

An intersecting family of sets is trivial if all of its members share a common element. Hilton and Milner proved a strong stability result for the celebrated Erdős--Ko--Rado theorem: when $n> 2k$, every non-trivial intersecting family of $k$-subsets of $[n]$ has at most $\binom{n-1}{k-1}-\binom{n-k-1}{k-1}+1$ members. One extremal family $\mathcal{HM}_{n, k}$ consists of a $k$-set $S$ and all $k$-subsets of $[n]$ containing a fixed element $x\not\in S$ and at least one element of $S$. We prove a degree version of the Hilton--Milner theorem: if $n=Ω(k^2)$ and $\mathcal{F}$ is a non-trivial intersecting family of $k$-subsets of $[n]$, then $δ(\mathcal{F})\le δ(\mathcal{HM}_{n.k})$, where $δ(\mathcal{F})$ denotes the minimum (vertex) degree of $\mathcal{F}$. Our proof uses several fundamental results in extremal set theory, the concept of kernels, and a new variant of the Erdős--Ko--Rado theorem.

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arXiv2017

A size-sensitive inequality for cross-intersecting families

Two families $\mathcal A$ and $\mathcal B$ of $k$-subsets of an $n$-set are called cross-intersecting if $A\cap B\ne\emptyset$ for all $A\in \mathcal A, B\in \mathcal B $. Strengthening the classical Erd\H os-Ko-Rado theorem, Pyber proved that $|\mathcal A||\mathcal B|\le {n-1\choose k-1}^2$ holds for $n\ge 2k$. In the present paper we sharpen this inequality. We prove that assuming $|\mathcal B|\ge {n-1\choose k-1}+{n-i\choose k-i+1}$ for some $3\le i\le k+1$ the stronger inequality $$|\mathcal A||\mathcal B|\le \Bigl({n-1\choose k-1}+{n-i\choose k-i+1}\Bigr)\Bigl({n-1\choose k-1}-{n-i\choose k-1}\Bigr)$$ holds. These inequalities are best possible.

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arXiv2017

Uniform s-cross-intersecting families

In this paper we study a question related to the classical Erdős-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 ${n-1\choose k-1}$. We say that two non-empty families are $\mathcal A, \mathcal B\subset {[n]\choose k}$ are {\it $s$-cross-intersecting}, if for any $A\in\mathcal A,B\in \mathcal B$ we have $|A\cap B|\ge s$. In this paper we determine the maximum of $|\mathcal A|+|\mathcal B|$ for all $n$. This generalizes a result of Hilton and Milner, who determined the maximum of $|\mathcal A|+|\mathcal B|$ for nonempty $1$-cross-intersecting families.

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arXiv2016

Short Proof of Erd\H os Conjecture for Triple Systems

In 1965 Erd\H os conjectured that for all $k\ge2$, $s\ge1$ and $n\ge k(s+1)$, an $n$-vertex $k$-uniform hypergraph $\F$ with $ν(\F)=s$ cannot have more than \newline $\max\{\binom{sk+k-1}k,\;\binom nk-\binom{n-s}k\}$ edges. It took almost fifty years to prove it for triple systems. In 2012 we proved the conjecture for all $s$ and all $n\ge4(s+1)$. Then Łuczak and Mieczkowska (2013) proved the conjecture for sufficiently large $s$ and all $n$. Soon after, Frankl proved it for all $s$. Here we present a simpler version of that proof which yields Erd\H os's conjecture for $s\ge33$. Our motivation is to lay down foundations for a possible proof in the much harder case $k=4$, at least for large $s$.

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arXiv2016

Multiply union families in $\mathbb{N}^n$

Let $A\subset \mathbb{N}^{n}$ be an $r$-wise $s$-union family, that is, a family of sequences with $n$ components of non-negative integers such that for any $r$ sequences in $A$ the total sum of the maximum of each component in those sequences is at most $s$. We determine the maximum size of $A$ and its unique extremal configuration provided (i) $n$ is sufficiently large for fixed $r$ and $s$, or (ii) $n=r+1$.

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arXiv2015

Decomposition of a cube into nearly equal smaller cubes

Let $d$ be a fixed positive integer and let $ε>0$. It is shown that for every sufficiently large $n\geq n_0(d,ε)$, the $d$-dimensional unit cube can be decomposed into exactly $n$ smaller cubes such that the ratio of the side length of the largest cube to the side length of the smallest one is at most $1+ε$. Moreover, for every $n\geq n_0$, there is a decomposition with the required properties, using cubes of at most $d+2$ different side lengths. If we drop the condition that the side lengths of the cubes must be roughly equal, it is sufficient to use cubes of two different sizes.

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arXiv2014

An Erd\H os--Ko--Rado theorem for cross $t$-intersecting families

Two families $\mathcal{A}$ and $\mathcal{B}$, of $k$-subsets of an $n$-set, are {\em cross $t$-intersecting} if for every choice of subsets $A \in \mathcal{A}$ and $B \in \mathcal{B}$ we have $|A \cap B| \geq t$. We address the following conjectured cross $t$-intersecting version of the Erd\H os--Ko--Rado Theorem: For all $n \geq (t+1)(k-t+1)$ the maximum value of $|\mathcal{A}||\mathcal{B}|$ for two cross $t$-intersecting families $\mathcal{A}, \mathcal{B} \subset\binom{[n]}{k}$ is $\binom{n-t}{k-t}^2$. We verify this for all $t \geq 14$ except finitely many $n$ and $k$ for each fixed $t$. Further, we prove uniqueness and stability results in these cases, showing, for instance, that the families reaching this bound are unique up to isomorphism. We also consider a {\em $p$-weight} version of the problem, which comes from the product measure on the power set of an $n$-set.

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arXiv2012

Large matchings in uniform hypergraphs and the conjectures of Erdos and Samuels

In this paper we study conditions which guarantee the existence of perfect matchings and perfect fractional matchings in uniform hypergraphs. We reduce this problem to an old conjecture by Erdős on estimating the maximum number of edges in a hypergraph when the (fractional) matching number is given, which we are able to solve in some special cases using probabilistic techniques. Based on these results, we obtain some general theorems on the minimum $d$-degree ensuring the existence of perfect (fractional) matchings. In particular, we asymptotically determine the minimum vertex degree which guarantees a perfect matching in 4-uniform and 5-uniform hypergraphs. We also discuss an application to a problem of finding an optimal data allocation in a distributed storage system.

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arXiv2011

A new short proof of the EKR theorem

A family F is intersecting if any two members have a nonempty intersection. Erdos, Ko, and Rado showed that |F|\leq {n-1\choose k-1} holds for an intersecting family of k-subsets of [n]:={1,2,3,...,n}, n\geq 2k. For n> 2k the only extremal family consists of all k-subsets containing a fixed element. Here a new proof is presented. It is even shorter than the classical proof of Katona using cyclic permutations, or the one found by Daykin applying the Kruskal-Katona theorem.

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