scholarly journals Two-Part Set Systems

10.37236/2067 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Péter L. Erdős ◽  
Dániel Gerbner ◽  
Nathan Lemons ◽  
Dhruv Mubayi ◽  
Cory Palmer ◽  
...  
Keyword(s):  
Intersection Property ◽  
Independent Interest ◽  
Sperner Property ◽  
Set Systems ◽  
Intersecting Families ◽  
Element Set ◽  
The Way ◽  
Families Of Subsets

The two part Sperner theorem of Katona and Kleitman states that if $X$ is an $n$-element set with partition $X_1 \cup X_2$, and $\mathcal{F}$ is a family of subsets of $X$ such that  no two sets $A, B \in \mathcal{F}$  satisfy $A \subset B$ (or $B \subset A$) and $A \cap X_i=B\cap X_i$ for some $i$, then $|\mathcal{F}| \le {n \choose \lfloor n/2\rfloor}$. We consider variations of this problem by replacing the Sperner property with the intersection property and considering families that satisfy various combinations of these properties on one or both parts $X_1$, $X_2$. Along the way, we prove the following  new result which may be of independent interest: let $\mathcal{F},\mathcal{G}$ be intersecting families of subsets of an $n$-element set that are additionally cross-Sperner, meaning that if $A \in\mathcal{F}$ and $B \in \mathcal{G}$, then $A \not\subset B$ and $B \not\subset A$. Then  $|\mathcal{F}| +|\mathcal{G}| \le 2^{n-1}$ and there are exponentially many examples showing that this bound is tight.

scholarly journals A New Upper Bound for Cancellative Pairs

10.37236/7210 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Barnabás Janzer
Keyword(s):  
Upper Bound ◽  
Set Systems ◽  
Element Set ◽  
Families Of Subsets

A pair $(\mathcal{A},\mathcal{B})$ of families of subsets of an $n$-element set is called cancellative if whenever $A,A'\in\mathcal{A}$ and $B\in\mathcal{B}$ satisfy $A\cup B=A'\cup B$, then $A=A'$, and whenever $A\in\mathcal{A}$ and $B,B'\in\mathcal{B}$ satisfy $A\cup B=A\cup B'$, then $B=B'$. It is known that there exist cancellative pairs with $|\mathcal{A}||\mathcal{B}|$ about $2.25^n$, whereas the best known upper bound on this quantity is $2.3264^n$. In this paper we improve this upper bound to $2.2682^n$. Our result also improves the best known upper bound for Simonyi's sandglass conjecture for set systems.

Flat functors and classifying toposes

2018 ◽  
Author(s):  
Olivia Caramello
Keyword(s):  
General Theory ◽  
Geometric Theory ◽  
Small Category ◽  
Independent Interest ◽  
Yoneda Lemma ◽  
Geometric Morphisms ◽  
The Way

This chapter develops a general theory of extensions of flat functors along geometric morphisms of toposes; the attention is focused in particular on geometric morphisms between presheaf toposes induced by embeddings of categories and on geometric morphisms to the classifying topos of a geometric theory induced by a small category of set-based models of the latter. A number of general results of independent interest are established on the way, including developments on colimits of internal diagrams in toposes and a way of representing flat functors by using a suitable internalized version of the Yoneda lemma. These general results will be instrumental for establishing in Chapter 6 the main theorem characterizing the class of geometric theories classified by a presheaf topos and for applying it.

scholarly journals Rational points on linear slices of diagonal hypersurfaces

2015 ◽  
Vol 218 ◽  
pp. 51-100
Author(s):  
Jörg Brüdern ◽  
Olivier Robert
Keyword(s):  
Asymptotic Formula ◽  
Rational Points ◽  
Independent Interest ◽  
The Way ◽  
Hardy Littlewood Method

AbstractAn asymptotic formula is obtained for the number of rational points of bounded height on the class of varieties described in the title line. The formula is proved via the Hardy-Littlewood method, and along the way we establish two new results on Weyl sums that are of some independent interest.

scholarly journals Cross L-intersecting families on set systems

2010 ◽  
Vol 310 (4) ◽  
pp. 720-726 ◽  
Author(s):  
Jiuqiang Liu ◽  
Xiaodong Liu
Keyword(s):  
Set Systems ◽  
Intersecting Families

scholarly journals On the union of intersecting families

2019 ◽  
Vol 28 (06) ◽  
pp. 826-839
Author(s):  
David Ellis ◽  
Noam Lifshitz
Keyword(s):  
Fixed Integer ◽  
Empty Intersection ◽  
Intersecting Family ◽  
The Family ◽  
Intersecting Families ◽  
Element Set

AbstractA family of sets is said to be intersecting if any two sets in the family have non-empty intersection. In 1973, Erdős raised the problem of determining the maximum possible size of a union of r different intersecting families of k-element subsets of an n-element set, for each triple of integers (n, k, r). We make progress on this problem, proving that for any fixed integer r ⩾ 2 and for any $$k \le ({1 \over 2} - o(1))n$$, if X is an n-element set, and $${\cal F} = {\cal F}_1 \cup {\cal F}_2 \cup \cdots \cup {\cal F}_r $$, where each $$ {\cal F}_i $$ is an intersecting family of k-element subsets of X, then $$|{\cal F}| \le \left( {\matrix{n \cr k \cr } } \right) - \left( {\matrix{{n - r} \cr k \cr } } \right)$$, with equality only if $${\cal F} = \{ S \subset X:|S| = k,\;S \cap R \ne \emptyset \} $$ for some R ⊂ X with |R| = r. This is best possible up to the size of the o(1) term, and improves a 1987 result of Frankl and Füredi, who obtained the same conclusion under the stronger hypothesis $$k < (3 - \sqrt 5 )n/2$$, in the case r = 2. Our proof utilizes an isoperimetric, influence-based method recently developed by Keller and the authors.

scholarly journals Anticlusters and intersecting families of subsets

1989 ◽  
Vol 51 (1) ◽  
pp. 90-103 ◽  
Author(s):  
Jerrold R Griggs ◽  
James W Walker
Keyword(s):  
Intersecting Families ◽  
Families Of Subsets

scholarly journals Reputable List Curation from Decentralized Voting

2020 ◽  
Vol 2020 (4) ◽  
pp. 297-320
Author(s):  
Elizabeth C. Crites ◽  
Mary Maller ◽  
Sarah Meiklejohn ◽  
Rebekah Mercer
Keyword(s):  
Real World ◽  
Independent Interest ◽  
Voting Scheme ◽  
World Information ◽  
The Way

AbstractToken-curated registries (TCRs) are a mechanism by which a set of users are able to jointly curate a reputable list about real-world information. Entries in the registry may have any form, so this primitive has been proposed for use—and deployed—in a variety of decentralized applications, ranging from the simple joint creation of lists to helping to prevent the spread of misinformation online. Despite this interest, the security of this primitive is not well understood, and indeed existing constructions do not achieve strong or provable notions of security or privacy. In this paper, we provide a formal cryptographic treatment of TCRs as well as a construction that provably hides the votes cast by individual curators. Along the way, we provide a model and proof of security for an underlying voting scheme, which may be of independent interest. We also demonstrate, via an implementation and evaluation, that our construction is practical enough to be deployed even on a constrained decentralized platform like Ethereum.

scholarly journals Stability Analysis for $k$-wise Intersecting Families

10.37236/602 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Vikram Kamat
Keyword(s):  
Stability Analysis ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Circle Method ◽  
Intersecting Family ◽  
Intersecting Families ◽  
Element Set

We consider the following generalization of the seminal Erdős–Ko–Rado theorem, due to Frankl. For some $k\geq 2$, let $\mathcal{F}$ be a $k$-wise intersecting family of $r$-subsets of an $n$ element set $X$, i.e. for any $F_1,\ldots,F_k\in \mathcal{F}$, $\cap_{i=1}^k F_i\neq \emptyset$. If $r\leq \dfrac{(k-1)n}{k}$, then $|\mathcal{F}|\leq {n-1 \choose r-1}$. We prove a stability version of this theorem, analogous to similar results of Dinur-Friedgut, Keevash-Mubayi and others for the EKR theorem. The technique we use is a generalization of Katona's circle method, initially employed by Keevash, which uses expansion properties of a particular Cayley graph of the symmetric group.

scholarly journals Set-Systems with Restricted Multiple Intersections

10.37236/1625 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Vince Grolmusz
Keyword(s):  
Upper Bound ◽  
Explicit Construction ◽  
Partial Answer ◽  
Residue Classes ◽  
Set Systems ◽  
Element Set ◽  
Multiple Intersections

We give a generalization for the Deza-Frankl-Singhi Theorem in case of multiple intersections. More exactly, we prove, that if ${\cal H}$ is a set-system, which satisfies that for some $k$, the $k$-wise intersections occupy only $\ell$ residue-classes modulo a $p$ prime, while the sizes of the members of ${\cal H}$ are not in these residue classes, then the size of ${\cal H}$ is at most $$(k-1)\sum_{i=0}^{\ell}{n\choose i}$$ This result considerably strengthens an upper bound of Füredi (1983), and gives partial answer to a question of T. Sós (1976). As an application, we give a direct, explicit construction for coloring the $k$-subsets of an $n$ element set with $t$ colors, such that no monochromatic complete hypergraph on $$\exp{(c(\log m)^{1/t}(\log \log m)^{1/(t-1)})}$$ vertices exists.

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