The disjointness graph G = G(𝒮) of a set of segments 𝒮 in ${\mathbb{R}^d}$ , $$d \ge 2$$ , is a graph whose vertex set is 𝒮 and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that... more
The disjointness graph G = G(𝒮) of a set of segments 𝒮 in ${\mathbb{R}^d}$ , $$d \ge 2$$ , is a graph whose vertex set is 𝒮 and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of G satisfies $\chi (G) \le {(\omega (G))^4} + {(\omega (G))^3}$ , where ω(G) denotes the clique number of G. It follows that 𝒮 has Ω(n1/5) pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments. We show that computing ω(G) and χ(G) for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colourings of G in which the number of colours satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free (ω(G) = 2), but whose chromatic numbers are arbitrarily...
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The graph obtained from the integer grid $\mathbb{Z}\times\mathbb{Z}$ by the removal of all horizontal edges that do not belong to the $x$-axis is called a comb. In a random walk on a graph, whenever a walker is at a vertex $v$, in the... more
The graph obtained from the integer grid $\mathbb{Z}\times\mathbb{Z}$ by the removal of all horizontal edges that do not belong to the $x$-axis is called a comb. In a random walk on a graph, whenever a walker is at a vertex $v$, in the next step it will visit one of the neighbors of $v$, each with probability $1/d(v)$, where $d(v)$ denotes the degree of $v$. We answer a question of Csáki, Csörgő, Földes, Révész, and Tusnády by showing that the expected number of vertices visited by a random walk on the comb after $n$ steps is $(\frac1{2\sqrt{2\pi}}+o(1))\sqrt{n}\log n.$ This contradicts a claim of Weiss and Havlin.
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Given any positive integers $m$ and $d$, we say a sequence of points $(x_{i})_{i\in I}$ in $\mathbb{R}^{m}$ is Lipschitz-$d$-controlling if one can select suitable values $y_{i}\;(i\in I)$ such that for every Lipschitz function... more
Given any positive integers $m$ and $d$, we say a sequence of points $(x_{i})_{i\in I}$ in $\mathbb{R}^{m}$ is Lipschitz-$d$-controlling if one can select suitable values $y_{i}\;(i\in I)$ such that for every Lipschitz function $f\,:\,\mathbb{R}^{m}\,\rightarrow \,\mathbb{R}^{d}$ there exists $i$ with $|f(x_{i})\,-\,y_{i}|\,<\,1$. We conjecture that for every $m\leqslant d$, a sequence $(x_{i})_{i\in I}\subset \mathbb{R}^{m}$ is $d$-controlling if and only if $$\begin{eqnarray}\displaystyle \sup _{n\in \mathbb{N}}\frac{|\{i\in I:|x_{i}|\leqslant n\}|}{n^{d}}=\infty . & & \displaystyle \nonumber\end{eqnarray}$$ We prove that this condition is necessary and a slightly stronger one is already sufficient for the sequence to be $d$-controlling. We also prove the conjecture for $m=1$.
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Consider the set H of all linear (or affine) transformations between two vector spaces over a finite field F. We study how good H is as a class of hash functions, namely we consider hashing a set S of size<br />n into a range having... more
Consider the set H of all linear (or affine) transformations between two vector spaces over a finite field F. We study how good H is as a class of hash functions, namely we consider hashing a set S of size<br />n into a range having the same cardinality n by a randomly chosen function from H and look at the expected size of the largest hash bucket. H is a universal class of hash functions for any finite field, but<br />with respect to our measure different fields behave differently.
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At most how many edges (hyperedges, nonzero entries, characters) can a graph (hypergraph, zero-one matrix, string) have if it does not contain a fixed forbidden pattern? Turán-type extremal graph theory, Erdős--Ko--Rado-type extremal set... more
At most how many edges (hyperedges, nonzero entries, characters) can a graph (hypergraph, zero-one matrix, string) have if it does not contain a fixed forbidden pattern? Turán-type extremal graph theory, Erdős--Ko--Rado-type extremal set theory, Ramsey theory, the theory of Davenport--Schinzel sequences, etc. have been developed to address questions of this kind. They produced a number of results that found important
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... from his copy before distributing illegal copies. To prevent such fraud, it is natural to distribute the digits of the fingerprint into locations of the digital document that are unknown to the users. The digits in these positions... more
... from his copy before distributing illegal copies. To prevent such fraud, it is natural to distribute the digits of the fingerprint into locations of the digital document that are unknown to the users. The digits in these positions must be ...
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We prove that for every k&amp;amp;amp;gt; 1, there exist k-fold coverings of the plane (1) with strips,(2) with axis-parallel rectangles, and (3) with homothets of any fixed concave quadrilateral, that cannot be decomposed into two... more
We prove that for every k&amp;amp;amp;gt; 1, there exist k-fold coverings of the plane (1) with strips,(2) with axis-parallel rectangles, and (3) with homothets of any fixed concave quadrilateral, that cannot be decomposed into two coverings. We also construct, for every k&amp;amp;amp;gt; 1, a set of ...
We find tight estimates for the minimum number of proper subspaces needed to cover all lattice points in an n-dimensional convex body symmetric about the origin. We also find the order of magnitude of the number of (n-1)-dimensional... more
We find tight estimates for the minimum number of proper subspaces needed to cover all lattice points in an n-dimensional convex body symmetric about the origin. We also find the order of magnitude of the number of (n-1)-dimensional subspaces induced by the lattice points in a large n-dimensional ball centered at the origin.
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Consider the set H of all linear (or affine) transformations between two vector spaces over a finite field F. We study how good H is as a class of hash functions, namely we consider hashing a set S of size n into a range having the same... more
Consider the set H of all linear (or affine) transformations between two vector spaces over a finite field F. We study how good H is as a class of hash functions, namely we consider hashing a set S of size n into a range having the same cardinality n by a randomly chosen function from H and look at the
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We construct binary codes for fingerprinting digital documents. Our codes for n users that are &amp;amp;amp;epsi;-secure against c pirates have length O(c2log(n/&amp;amp;amp;epsi;)). This improves the codes proposed by Boneh and... more
We construct binary codes for fingerprinting digital documents. Our codes for n users that are &amp;amp;amp;epsi;-secure against c pirates have length O(c2log(n/&amp;amp;amp;epsi;)). This improves the codes proposed by Boneh and Shaw &amp;amp;amp;lsqb;1998&amp;amp;amp;rsqb; whose length is approximately the square of this length. The improvement carries over to works using the Boneh--Shaw code as a primitive, for example, to the dynamic traitor
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Answering a question of Alon, Ding, Oporowski and Vertigan (4), we show that there exists an absolute constant C such that every graph G with maximum degree 5 has a vertex partition into 2 parts, such that the subgraph induced by each... more
Answering a question of Alon, Ding, Oporowski and Vertigan (4), we show that there exists an absolute constant C such that every graph G with maximum degree 5 has a vertex partition into 2 parts, such that the subgraph induced by each part has no component of size greater than C. We obtain similar results for partitioning graphs of given
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Let G be a geometric graph with n vertices, ie, a graph drawn in the plane with straight-line edges. It is shown that if G has no self-intersecting path of length 3, then its number of edges is O (n log n). This result is asymptotically... more
Let G be a geometric graph with n vertices, ie, a graph drawn in the plane with straight-line edges. It is shown that if G has no self-intersecting path of length 3, then its number of edges is O (n log n). This result is asymptotically tight. Analogous questions for curvilinear ...