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pro vyhledávání: '"Frankl, A."'
We study a variant of the Erd\H{o}s Matching Problem in random hypergraphs. Let $\mathcal{K}_p(n,k)$ denote the Erd\H{o}s-R\'enyi random $k$-uniform hypergraph on $n$ vertices where each possible edge is included with probability $p$. We show that wh
Externí odkaz:
http://arxiv.org/abs/2410.15585
Autor:
Frankl, Peter, Wang, Jian
We consider families, $\mathcal{F}$ of $k$-subsets of an $n$-set. For integers $r\geq 2$, $t\geq 1$, $\mathcal{F}$ is called $r$-wise $t$-intersecting if any $r$ of its members have at least $t$ elements in common. The most natural construction of su
Externí odkaz:
http://arxiv.org/abs/2409.19344
Autor:
Balko, Martin, Frankl, Nóra
The celebrated Szemer\'edi--Trotter theorem states that the maximum number of incidences between $n$ points and $n$ lines in the plane is $O(n^{4/3})$, which is asymptotically tight. Solymosi (2005) conjectured that for any set of points $P_0$ and fo
Externí odkaz:
http://arxiv.org/abs/2409.00954
Autor:
Buchanan, Calum, Clifton, Alexander, Culver, Eric, Frankl, Péter, Nie, Jiaxi, Ozeki, Kenta, Rombach, Puck, Yin, Mei
Babai and Frankl posed the ``odd cover problem" of finding the minimum cardinality of a collection of complete bipartite graphs such that every edge of the complete graph of order $n$ is covered an odd number of times. In a previous paper with O'Neil
Externí odkaz:
http://arxiv.org/abs/2408.08598
Autor:
Frankl, Peter, Kupavskii, Andrey
Let us consider a collection $\mathcal G$ of codewords of length $n$ over an alphabet of size $s$. Let $t_1,\ldots, t_s$ be nonnegative integers. What is the maximum of $|\mathcal G|$ subject to the condition that any two codewords should have at lea
Externí odkaz:
http://arxiv.org/abs/2408.08221
Autor:
Frankl, Peter, Wang, Jian
A $k$-uniform family $\mathcal{F}$ is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. The shadow family $\partial \mathcal{F}$ is the family of $(k-1)$-element sets that are contained in some members of $\mathcal{F}$. T
Externí odkaz:
http://arxiv.org/abs/2406.00465
There are four non-isomorphic configurations of triples that can form a triangle in a $3$-uniform hypergraph. Forbidding different combinations of these four configurations, fifteen extremal problems can be defined, several of which already appeared
Externí odkaz:
http://arxiv.org/abs/2405.16452
Autor:
Frankl, Peter, Kupavskii, Andrey
A family of sets is $r$-wise agreeing if for any $r$ sets from the family there is an element $x$ that is either contained in all or contained in none of the $r$ sets. The study of such families is motivated by questions in discrete optimization. In
Externí odkaz:
http://arxiv.org/abs/2404.14178
Two celebrated extensions of Helly's theorem are the Fractional Helly theorem of Katchalski and Liu (1979) and the Quantitative Volume theorem of B\'ar\'any, Katchalski, and Pach (1982). Improving on several recent works, we prove an optimal combinat
Externí odkaz:
http://arxiv.org/abs/2402.12268
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 of them. It follows from the Erd\H os--Szemer\'edi conjecture, recently
Externí odkaz:
http://arxiv.org/abs/2310.16701