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The ultimate mathematics reference bookThis is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries—written especially for thi
In its usual form, Freiman's 3 k − 4 $3k-4$ theorem states that if A $A$ and B $B$ are subsets of Z ${\mathbb {Z}}$ of size k $k$ with small sumset (of size close to 2 k $2k$ ), then they are very close to arithmetic progressions. Our aim in this p
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_dedup___::ed983e9d95ed27e9a37b87d870870fba
https://www.repository.cam.ac.uk/handle/1810/350321
https://www.repository.cam.ac.uk/handle/1810/350321
A (finite or infinite) graph is called constructible if it may be obtained recursively from the one-point graph by repeatedly adding dominated vertices. In the finite case, the constructible graphs are precisely the cop-win graphs, but for infinite g
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_dedup___::458ce6ed451d33c20373318505708712
Publikováno v:
European Journal of Combinatorics. 79:46-59
How large an antichain can we find inside a given downset in the lattice of subsets of [n]? Sperner's theorem asserts that the largest antichain in the whole lattice has size the binomial coefficient C(n, n/2); what happens for general downsets? Our
A factorisation $x = u_1 u_2 \cdots$ of an infinite word $x$ on alphabet $X$ is called `monochromatic', for a given colouring of the finite words $X^*$ on alphabet $X$, if each $u_i$ is the same colour. Wojcik and Zamboni proved that the word $x$ is
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_dedup___::89d3f2c850b2d4b0c1f154f0c82fbfd2
http://arxiv.org/abs/2010.09081
http://arxiv.org/abs/2010.09081
Autor:
Paul A. Russell, Imre Leader
Publikováno v:
The Electronic Journal of Combinatorics. 27
We say that the system of equations $Ax = b$, where $A$ is an integer matrix and $b$ is a (non-zero) integer vector, is partition regular if whenever the integers are finitely coloured there is a monochromatic vector $x$ with $Ax = b.$ Rado proved th
We show that the Union-Closed Conjecture holds for the union-closed family generated by the cyclic translates of any fixed set.
Comment: 3 pages. Updated references
Comment: 3 pages. Updated references
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_dedup___::add4b57479b3e55f4d62380716e3aa25
We show that there exists an absolute constant $C>0$ such that any family $\mathcal{F}\subset \{0,1\}^n$ of size at least $Cn^3$ has dual VC-dimension at least 3. Equivalently, every family of size at least $Cn^3$ contains three sets such that all ei
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_dedup___::bda81d0e1111436e29f628bfd47a5c60
http://arxiv.org/abs/1911.00487
http://arxiv.org/abs/1911.00487
Publikováno v:
Balister, P, Bollobás, B, Gunderson, K, Leader, I & Walters, M 2015, ' Random Geometric Graphs and Isometries of Normed Spaces ', arXiv .
Given a countable dense subset S S of a finite-dimensional normed space X X , and 0 > p > 1 0>p>1 , we form a random graph on S S by joining, independently and with probability p p , each pair of points at distance less than 1 1 . We say that S S is
Publikováno v:
Electronic Notes in Discrete Mathematics. 61:535-539
Let P be a partially ordered set with a unique maximal and minimal element, and size 2m, where m is a positive integer. Settling a conjecture of Lonc, we prove that if n is sufficiently large, then the Boolean lattice 2[n] can be partitioned into iso