arXiv subjects

Andrey Kupavskii

Publications and source records attributed to Andrey Kupavskii.

118 records · Page 7Linked 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.

math.CO↗

arXiv2017

On the size of $k$-cross-free families

Two subsets $A,B$ of an $n$-element ground set $X$ are said to be \emph{crossing}, if none of the four sets $A\cap B$, $A\setminus B$, $B\setminus A$ and $X\setminus(A\cup B)$ are empty. It was conjectured by Karzanov and Lomonosov forty years ago that if a family $\mathcal{F}$ of subsets of $X$ does not contain $k$ pairwise crossing elements, then $|\mathcal{F}|=O_{k}(n)$. For $k=2$ and $3$, the conjecture is true, but for larger values of $k$ the best known upper bound, due to Lomonosov, is $|\mathcal{F}|=O_{k}(n\log n)$. In this paper, we improve this bound by showing that $|\mathcal{F}|=O_{k}(n\log^{*} n)$ holds, where $\log^{*}$ denotes the iterated logarithm function.

math.CO↗

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.

math.CO↗

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.

math.CO↗

arXiv2016

Number of double-normal pairs in space

Given a set $V$ of points in $\mathbb R^d$, two points $p$, $q$ from $V$ form a double-normal pair, if the set $V$ lies between two parallel hyperplanes that pass through $p$ and $q$, respectively, and that are orthogonal to the segment $pq$. In this paper we study the maximum number $N_d(n)$ of double-normal pairs in a set of $n$ points in $\mathbb R^d$. It is not difficult to get from the famous Erdős-Stone theorem that $N_d(n) = \frac 12(1-1/k)n^2+o(n^2)$ for a suitable integer $k = k(d)$ and it was shown in the paper by J. Pach and K. Swanepoel that $\lceil d/2\rceil\le k(d)\le d-1$ and that asymptotically $k(d)\gtrsim d-O(\log d)$. In this paper we sharpen the upper bound on $k(d)$, which, in particular, gives $k(4)=2$ and $k(5)=3$ in addition to the equality $k(3)=2$ established by J. Pach and K. Swanepoel. Asymptotically we get $k(d)\le d- \log_2k(d) = d - (1+ o(1)) \log_2k(d)$ and show that this problem is connected with the problem of determining the maximum number of points in $\mathbb R^d$ that form pairwise acute (or non-obtuse) angles.

math.CO↗

arXiv2014

Diameter graphs in $\mathbb R^4$

A \textit{diameter graph in $\mathbb R^d$} is a graph, whose set of vertices is a finite subset of $\mathbb R^d$ and whose set of edges is formed by pairs of vertices that are at diameter apart. This paper is devoted to the study of different extremal properties of diameter graphs in $\mathbb R^4$ and on a three-dimensional sphere. We prove an analogue of Vázsonyi's and Borsuk's conjecture for diameter graphs on a three-dimensional sphere with radius greater than $1/\sqrt 2$. We prove Schur's conjecture for diameter graphs in $\mathbb R^4.$ We also establish the maximum number of triangles a diameter graph in $\mathbb R^4$ can have, showing that the extremum is attained only on specific Lenz configurations.

math.CO↗

arXiv2014

Two notions of unit distance graphs

A {\em faithful (unit) distance graph} in $\mathbb{R}^d$ is a graph whose set of vertices is a finite subset of the $d$-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly $1$. A {\em (unit) distance graph} in $\mathbb{R}^d$ is any subgraph of such a graph. In the first part of the paper we focus on the differences between these two classes of graphs. In particular, we show that for any fixed $d$ the number of faithful distance graphs in $\mathbb{R}^d$ on $n$ labelled vertices is $2^{(1+o(1)) d n \log_2 n}$, and give a short proof of the known fact that the number of distance graphs in $\mathbb{R}^d$ on $n$ labelled vertices is $2^{(1-1/\lfloor d/2 \rfloor +o(1))n^2/2}$. We also study the behavior of several Ramsey-type quantities involving these graphs. % and high-girth graphs from these classes. In the second part of the paper we discuss the problem of determining the minimum possible number of edges of a graph which is not isomorphic to a faithful distance graph in $\mathbb R^d$.

math.CO↗

arXiv2013

New bounds for the distance Ramsey number

In this paper we study the distance Ramsey number $R_{\it D}(s,t,d)$. The \textit{distance Ramsey number} $R_{\it D}(s,t,d) $ is the minimum number $n$ such that for any graph $ G $ on $ n $ vertices, either $G$ contains an induced $ s $-vertex subgraph isomorphic to a distance graph in $ \Real^d $ or $ \bar {G} $ contains an induced $ t $-vertex subgraph isomorphic to the distance graph in $ \Real^d $. We obtain the upper and lower bounds on $R_{\it D}(s,s,d),$ which are similar to the bounds for the classical Ramsey number $R(\lceil \frac{s}{[d/2]} \rceil, \lceil \frac{s}{[d/2]} \rceil)$.

math.CO↗

arXiv2013

Distance graphs with large chromatic number and arbitrary girth

In this article we consider a problem related to two famous combinatorial topics. One of them concerns the chromatic number of the space. The other deals with graphs having big girth (the length of the shortest cycle) and large chromatic number. Namely, we prove that for any $l\in \mathbb{N}$ there exists a sequence of distance graphs in $\mathbb{R}^n$ with girth at least $l$ and the chromatic number equal to $(c+\bar{o}(1))^n$ with $c>1$.

math.CO↗

arXiv2013

The distribution of second degrees in the Buckley-Osthus random graph model

In this paper we consider a well-known generalization of the Barabási and Albert preferential attachment model - the Buckley-Osthus model. Buckley and Osthus proved that in this model the degree sequence has a power law distribution. As a natural (and arguably more interesting) next step, we study the second degrees of vertices. Roughly speaking, the second degree of a vertex is the number of vertices at distance two from this vertex. The distribution of second degrees is of interest because it is a good approximation of PageRank, where the importance of a vertex is measured by taking into account the popularity of its neighbors. We prove that the second degrees also obey a power law. More precisely, we estimate the expectation of the number of vertices with the second degree greater than or equal to k and prove the concentration of this random variable around its expectation using the now-famous Talagrand's concentration inequality over product spaces. As far as we know this is the only application of Talagrand's inequality to random web graphs, where the (preferential attachment) edges are not defined over a product distribution, making the application nontrivial, and requiring certain novelty.

math.PR↗