28
28

Sep 20, 2013
09/13

by
Peter Frankl

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The aim of the present paper is to prove that the maximum number of edges in a 3-uniform hypergraph on n vertices and matching number s is max{\binom(3s+2,3), \binom(n,3) - \binom(n-s,3)} for all n,s, n >= 3s+2.

Source: http://arxiv.org/abs/1205.6847v1

2
2.0

Jun 29, 2018
06/18

by
Peter Frankl; Andrey Kupavskii

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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...

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1603.00936

Track Listing: 01 Estampes - Pagodes 02 Estampes - Soirée dans Grenade 03 Estampes - Jardins sous la pluie 04 Ballade 05 Masques 06 Children's Corner - Dr. Gradus ad Parnassum 07 Children's Corner - Jimbo's Lullaby 08 Children's Corner - Serenade for the Doll 09 Children's Corner - Snow is Dancing 10 Children's Corner - The Little Shepherd 11 Children's Corner - Golliwog's Cake-walk 12 Berceuse héroique 13 Danse: Tarantelle Styrienne 14 Nocturne 15 Pour le piano - Prélude 16 Pour le piano -...

Topic: Classical

Source: CD

2
2.0

Jun 29, 2018
06/18

by
Peter Frankl; Andrey Kupavskii

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The families $\mathcal F_1,\ldots, \mathcal F_s\subset 2^{[n]}$ are called \textit{$q$-dependent} if there are no pairwise disjoint $F_i\in \mathcal F_i,\ i=1,\ldots, s,$ satisfying $|F_1\cup\ldots\cup F_s|\le q.$ We determine $\max |\mathcal F_1|+\ldots +|\mathcal F_s| $ for \textit{all} values $n\ge q,s\ge 2$. The result provides a far-reaching generalization of an important classical result of Kleitman. The uniform case $\mathcal F_1 = \ldots = \mathcal F_s\subset {[n]\choose k}$ of this...

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1607.06126

Source material did not include any artwork. Tracklist: 1. Andante & Variations For 2 Pianos, 2 Cellos & Horn In B Flat Major Op 46 (Original Version) 2. Kinderball, Op 130 (Sechs Leichte Tanzsucke) 3. Binder Aus Dem Osten, Op 66 (6 Impromptus)

Source: CD

2
2.0

Jun 30, 2018
06/18

by
Peter Frankl; Andrey Kupavskii

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In this paper we study a classical problem of extremal set theory, which asks for the maximum size of a family of subsets of $[n]$ such that $s$ sets 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...

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1701.04107

2
2.0

Jun 30, 2018
06/18

by
Peter Frankl; Andrey Kupavskii

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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 Erd\H os, 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 $n$ are there? Improving the result of Balogh et al., we determine the asymptotic of this quantity for $n\ge 2k+2+2\sqrt{k\log k}$ and...

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1701.04110

3
3.0

Jun 29, 2018
06/18

by
Peter Frankl; Andrey Kupavskii

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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 $k$ and $l$ and...

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1603.00938

2
2.0

Jun 28, 2018
06/18

by
Peter Frankl; Andrey Kupavskii

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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 $n=k^2$....

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1510.03912

110
110

Sep 21, 2013
09/13

by
Peter Frankl; Zoltan Furedi

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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.

Source: http://arxiv.org/abs/1108.2179v1

2
2.0

Jun 29, 2018
06/18

by
Peter Frankl; Andrey Kupavskii

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In this paper we study a question related to the classical Erd\H{o}s-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...

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1611.07258

2
2.0

Jun 28, 2018
06/18

by
Peter Frankl; Amram Meir; Janos Pach

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Let $d$ be a fixed positive integer and let $\epsilon>0$. It is shown that for every sufficiently large $n\geq n_0(d,\epsilon)$, 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 $1+\epsilon$. 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...

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1511.05301

Track Listings: Sonata for Violin & Piano no. 40 in B-flat : 1. I. Largo - allegro 2 II. Andante 3. III. Allegretto Violin Sonata in G: 4. I. Adagio 5. II. Allegro 6. III. Andantino cantabile Sonata for Violin & Piano no. 42 in A major: 7. I. Allegro molto 8. II. Andante 9. III. Presto

Topic: Classical

Source: CD

0
0.0

Jun 29, 2018
06/18

by
Peter Frankl; Vojtech Rödl; Andrzej Ruciński

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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 $\nu(\F)=s$ 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 {\L}uczak and Mieczkowska (2013) proved the conjecture for sufficiently large $s$ and all $n$. Soon after, Frankl proved it for all $s$. Here...

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1609.00530

0
0.0

Jun 28, 2018
06/18

by
Peter Frankl; Masashi Shinohara; Norihide Tokushige

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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) $n=r+1$.

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1511.03828

0
0.0

Jun 30, 2018
06/18

by
Peter Frankl; János Pach; Christian Reiher; Vojtěch Rödl

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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...

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1702.03707

27
27

Sep 23, 2013
09/13

by
Peter Frankl; Sang June Lee; Mark Siggers; Norihide Tokushige

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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...

Source: http://arxiv.org/abs/1303.0657v2

152
152

Jul 20, 2013
07/13

by
Noga Alon; Peter Frankl; Hao Huang; Vojtech Rodl; Andrzej Rucinski; Benny Sudakov

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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 Erd\H{o}s 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...

Source: http://arxiv.org/abs/1107.1219v2

Other Minds Music Reference Library

5
5.0

Oct 7, 2019
10/19

by
Beethoven; Dvořák; György Pauk; Ralph Kirshbaum; Peter Frankl; Antonín Dvořák; Ludwig van Beethoven

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Dirty disc inherent from the CD. Tracklist: 1. Piano Trio in B-flat, op. 97 ’Archduke’: Allegro moderato - Ludwig van Beethoven 2. Piano Trio in B-flat, op. 97 ’Archduke’: Scherzo. Allegro - Ludwig van Beethoven 3. Piano Trio in B-flat, op. 97 ’Archduke’: Andante cantabile - Ludwig van Beethoven 4. Piano Trio in B-flat, op. 97 ’Archduke’: Allegro moderato presto - Ludwig van Beethoven 5. Piano Trio in E minor, op. 90 ’Dumky’: Lento maestoso - Allegro vivace - Antonín...

Source: CD

14
14

Oct 9, 2014
10/14

by
Ludwig van Beethoven; Antonín Dvořák; György Pauk; Ralph Kirshbaum; Peter Frankl

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Track Listings: 1 Piano Trio in B Flat Op. 97 'Archduke': Allegro Moderato 2 Piano Trio in B Flat Op. 97 'Archduke': Scherzo, Allegro 3 Piano Trio in B Flat Op. 97 'Archduke': Andante Cantabile 4 Piano Trio in B Flat Op. 97 'Archduke': Allegro Moderato Presto 5 Piano Trio in E Minor, Op. 90 'Dumky': Lento Maestoso - Allegro Vivace 6 Piano Trio in E Minor, Op. 90 'Dumky': Poco Adagio - Vivace Non Troppo 7 Piano Trio in E Minor, Op. 90 'Dumky': Andante - Vivace Non Troppo 8 Piano Trio in E Minor,...

Topic: Other Classical

Source: CD