82 research outputs found
Hypergraph cuts above the average
An r-cut of a k-uniform hypergraph H is a partition of the vertex set of H into r parts and the size of the cut is the number of edges which have a vertex in each part. A classical result of Edwards says that every m-edge graph has a 2-cut of size m/2 + Ω(√m), and this is best possible. That is, there exist cuts which exceed the expected size of a random cut by some multiple of the standard deviation. We study analogues of this and related results in hypergraphs. First, we observe that similarly to graphs, every m-edge k-uniform hypergraph has an r-cut whose size is Ω(√m) larger than the expected size of a random r-cut. Moreover, in the case where k = 3 and r = 2 this bound is best possible and is attained by Steiner triple systems. Surprisingly, for all other cases (that is, if k ≥ 4 or r ≥ 3), we show that every m-edge k-uniform hypergraph has an r-cut whose size is Ω(m^(5/9)) larger than the expected size of a random r-cut. This is a significant difference in behaviour, since the amount by which the size of the largest cut exceeds the expected size of a random cut is now considerably larger than the standard deviation
On restricted colourings of Kn
The authors investigate Ramsey-type extremal problems for finite graphs. In Section 1, anti-Ramsey numbers for paths are determined. For positive integers k and n let r=f(n,Pk) be the maximal integer such that there exists an edge colouring of Kn using precisely r colours but not containing any coloured path on k vertices with all edges having different colors. It is shown that f(n,P2k+3+ε)=t⋅n−(t+12)+1+ε for t≥5, n>c⋅t2 and ε=0,1. In Section 2, K3-spectra of colourings are determined. Given S⊆{1,2,3}, the authors investigate for which r and n there exist edge colourings of Kn using precisely r colours such that all triangles are s-coloured for some s∈S and, conversely, every s∈S occurs. Section 3 contains suggestions for further research
Unavoidable Multicoloured Families of Configurations
Balogh and Bollob\'as [{\em Combinatorica 25, 2005}] prove that for any
there is a constant such that any set system with at least sets
reduces to a -star, an -costar or an -chain. They proved
. Here we improve it to for some constant
.
This is a special case of the following result on the multi-coloured
forbidden configurations at 2 colours. Let be given. Then there exists a
constant so that a matrix with entries drawn from with
at least different columns will have a submatrix that
can have its rows and columns permuted so that in the resulting matrix will be
either or (for some ), where
is the matrix with 's on the diagonal and 's else
where, the matrix with 's below the diagonal and
's elsewhere. We also extend to considering the bound on the number of
distinct columns, given that the number of rows is , when avoiding a matrix obtained by taking any one of the matrices above
and repeating each column times. We use Ramsey Theory.Comment: 16 pages, add two application
Anti-Ramsey theorems
Let c be an edge-colouring of the complete n-graph Kn with m colours. A totally multicoloured (TMC) subgraph of Kn (with respect to c) is a graph G⊆Kn such that no two edges of G have the same colour. Let H be a graph with less than n vertices. If m is sufficiently large then there is in Kn a TMC subgraph isomorphic to H. Let f(n,H) denote the maximal number m such that there is an m-colouring of Kn without a TMC-subgraph isomorphic to H. Put d=min(χ(H−e),e∈E(H))−1. It is shown that f(n,H)/(n2) converges to 1−1/d for n→∞. An analogous result is proved for uniform hypergraphs. If, especially, H is a complete graph Kp, the extremal colouring is, for every n, uniquely determined (up to isomorphisms); it is closely related to the Turán graph (with p−1 colour-classes). The problem of determining f(n,H) is also discussed in more detail in the case when H is a path or a circuit, and in the case of a path an explicit formula is given, holding for n sufficiently large
Point selections and weak e-nets for convex hulls
One of our results: let X be a finite set on the plane, 0 < g < 1, then there exists a set F (a weak g-net) of size at most 7/e 2 such that every convex set containing at least e\X\ elements of X intersects F. Note that the size of F is independent of the size of X. 1
Set Systems with Restricted Cross-Intersections and the Minimum Rank of Inclusion Matrices
A set system is L-intersecting if any pairwise intersection size lies in L, where L is some set of s nonnegative integers. The celebrated Frankl-Ray-Chaudhuri-Wilson theorems give tight bounds on the size of an L-intersecting set system on a ground set of size n. Such a system contains at most sets if it is uniform and at most sets if it is nonuniform. They also prove modular versions of these results.
We consider the following extension of these problems. Call the set systems {\em L-cross-intersecting} if for every pair of distinct sets A,B with and for some the intersection size lies in . For any k and for n > n 0 (s) we give tight bounds on the maximum of . It is at most if the systems are uniform and at most if they are nonuniform. We also obtain modular versions of these results.
Our proofs use tools from linear algebra together with some combinatorial ideas. A key ingredient is a tight lower bound for the rank of the inclusion matrix of a set system. The s*-inclusion matrix of a set system on [n] is a matrix M with rows indexed by and columns by the subsets of [n] of size at most s, where if and with , we define M AB to be 1 if and 0 otherwise. Our bound generalizes the well-known result that if , then M has full rank . In a combinatorial setting this fact was proved by Frankl and Pach in the study of null t-designs; it can also be viewed as determining the minimum distance of the Reed-Muller codes
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