A Ramsey-type result for the hypercube

2006 ◽  
Vol 53 (3) ◽  
pp. 196-208 ◽  
Author(s):  
Noga Alon ◽  
Radoš Radoičić ◽  
Benny Sudakov ◽  
Jan Vondrák
Keyword(s):  
Type Result ◽  
Ramsey Type

A RAMSEY TYPE RESULT FOR LATIN SQUARES

2019 ◽  
Vol 101 (3) ◽  
pp. 362-366
Author(s):  
YEVHEN ZELENYUK ◽  
YULIYA ZELENYUK
Keyword(s):  
Cartesian Product ◽  
Latin Square ◽  
Latin Squares ◽  
Multiplication Table ◽  
The Other ◽  
Type Result ◽  
Other Hand ◽  
Ramsey Type

We show that for all $m,k,r\in \mathbb{N}$, there is an $n\in \mathbb{N}$ such that whenever $L$ is a Latin square of order $m$ and the Cartesian product $L^{n}$ of $n$ copies of $L$ is $r$-coloured, there is a monochrome Latin subsquare of $L^{n}$, isotopic to $L^{k}$. In particular, for every prime $p$ and for all $k,r\in \mathbb{N}$, there is an $n\in \mathbb{N}$ such that whenever the multiplication table $L({\mathbb{Z}_{p}}^{n})$ of the group ${\mathbb{Z}_{p}}^{n}$ is $r$-coloured, there is a monochrome Latin subsquare of order $p^{k}$. On the other hand, we show that for every group $G$ of order $\leq 15$, there is a 2-colouring of $L(G)$ without a nontrivial monochrome Latin subsquare.

scholarly journals Applications of a New Separator Theorem for String Graphs

2013 ◽  
Vol 23 (1) ◽  
pp. 66-74 ◽  
Author(s):  
JACOB FOX ◽  
JÁNOS PACH
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Present Note ◽  
Intersection Graph ◽  
Type Result ◽  
Vertex Separator ◽  
String Graphs ◽  
String Graph ◽  
Ramsey Type ◽  
Separator Theorem

An intersection graph of curves in the plane is called astring graph. Matoušek almost completely settled a conjecture of the authors by showing that every string graph withmedges admits a vertex separator of size$O(\sqrt{m}\log m)$. In the present note, this bound is combined with a result of the authors, according to which every dense string graph contains a large complete balanced bipartite graph. Three applications are given concerning string graphsGwithnvertices: (i) ifKt⊈Gfor somet, then the chromatic number ofGis at most (logn)O(logt); (ii) ifKt,t⊈G, thenGhas at mostt(logt)O(1)nedges,; and (iii) a lopsided Ramsey-type result, which shows that the Erdős–Hajnal conjecture almost holds for string graphs.

scholarly journals Note on a Ramsey Theorem for Posets with Linear Extensions

10.37236/6392 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Andrii Arman ◽  
Vojtěch Rödl
Keyword(s):  
Short Proof ◽  
Partially Ordered Sets ◽  
Linear Extensions ◽  
Ordered Sets ◽  
Type Result ◽  
Ramsey Theorem ◽  
Partially Ordered ◽  
Ramsey Type

In this note we consider a Ramsey-type result for partially ordered sets. In particular, we give an alternative short proof of a theorem for a posets with multiple linear extensions recently obtained by Solecki and Zhao (2017).

Unit-distance graphs, graphs on the integer lattice and a Ramsey type result

1995 ◽  
Vol 49 (1) ◽  
pp. 319-319
Author(s):  
Kiran B. Chilakamarri ◽  
Carolyn R. Mahoney
Keyword(s):  
Integer Lattice ◽  
Type Result ◽  
Unit Distance ◽  
Distance Graphs ◽  
Ramsey Type

scholarly journals Large 2-Coloured Matchings in 3-Coloured Complete Hypergraphs

10.37236/2324 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Tamás Terpai
Keyword(s):  
Uniform Hypergraph ◽  
Type Result ◽  
Ramsey Type

We prove the generalized Ramsey-type result on large $2$-coloured matchings in a $3$-coloured complete $3$-uniform hypergraph, supporting a conjecture by A. Gyárfás.

scholarly journals A Ramsey-type result for geometricℓ-hypergraphs

2014 ◽  
Vol 41 ◽  
pp. 232-241 ◽  
Author(s):  
Dhruv Mubayi ◽  
Andrew Suk
Keyword(s):  
Type Result ◽  
Ramsey Type

Unit-distance graphs, graphs on the integer lattice and a Ramsey type result

1996 ◽  
Vol 51 (1-2) ◽  
pp. 48-67
Author(s):  
Kiran B. Chilakamarri ◽  
Carolyn R. Mahoney
Keyword(s):  
Integer Lattice ◽  
Type Result ◽  
Unit Distance ◽  
Distance Graphs ◽  
Ramsey Type

scholarly journals On the minimum size of a point set containing a 5-hole and a disjoint 4-hole

2011 ◽  
Vol 48 (4) ◽  
pp. 445-457 ◽  
Author(s):  
Bhaswar Bhattacharya ◽  
Sandip Das
Keyword(s):  
Upper Bound ◽  
Minimum Size ◽  
Convex Hulls ◽  
Type Result ◽  
Convex Polygons ◽  
Empty Convex ◽  
Ramsey Type ◽  
Point Set

Let H(k; l), k ≦ l denote the smallest integer such that any set of H(k; l) points in the plane, no three on a line, contains an empty convex k-gon and an empty convex l-gon, which are disjoint, that is, their convex hulls do not intersect. Hosono and Urabe [JCDCG, LNCS 3742, 117–122, 2004] proved that 12 ≦ H(4, 5) ≦ 14. Very recently, using a Ramseytype result for disjoint empty convex polygons proved by Aichholzer et al. [Graphs and Combinatorics, Vol. 23, 481–507, 2007], Hosono and Urabe [Kyoto CGGT, LNCS 4535, 90–100, 2008] improve the upper bound to 13. In this paper, with the help of the same Ramsey-type result, we prove that H(4; 5) = 12.

Extremal Problems for Affine Cubes of Integers

1998 ◽  
Vol 7 (1) ◽  
pp. 65-79 ◽  
Author(s):  
DAVID S. GUNDERSON ◽  
VOJTÉCH RÖDL
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Extremal Problems ◽  
Upper And Lower Bounds ◽  
Type Result ◽  
Ramsey Type ◽  
Positive Integers ◽  
Density Results

A collection H of integers is called an affine d-cube if there exist d+1 positive integers x0,x1,…, xd so thatformula hereWe address both density and Ramsey-type questions for affine d-cubes. Regarding density results, upper bounds are found for the size of the largest subset of {1,2,…,n} not containing an affine d-cube. In 1892 Hilbert published the first Ramsey-type result for affine d-cubes by showing that, for any positive integers r and d, there exists a least number n=h(d,r) so that, for any r-colouring of {1,2,…,n}, there is a monochromatic affine d-cube. Improvements for upper and lower bounds on h(d,r) are given for d>2.

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