Name
Abstract | ||
|---|---|---|
A set of containers for a hypergraph G is a collection C of vertex subsets, such that for every independent (or, indeed, merely sparse) set I of G there is some C ź C with I ź C , no member of C is large, and the collection C is relatively small. Containers with useful properties have been exhibited by Balogh, Morris and Samotij 6 and by the authors 39-41, along with several applications.Our purpose here is to give a simpler algorithm than the one used in 40, which nevertheless yields containers with all the properties needed for the main container theorem of 40 and its consequences. Moreover this algorithm produces containers having the so-called online property, allowing the colouring results of 40 to be extended to all, not just simple, hypergraphs. Most of the proof of the container theorem remains the same if this new algorithm is used, and we do not repeat all the details here, but describe only the changes that need to be made. However, for illustrative purposes, we do include a complete proof of a slightly weaker but simpler version of the theorem, which for many (perhaps most) applications is plenty.We also present applications to the number of solution-free sets of linear equations, including the number of Sidon sets, that were announced in 40. |
Year | DOI | Venue |
|---|---|---|
2016 | 10.1016/j.jctb.2016.05.011 | J. Comb. Theory, Ser. B |
Keywords | Field | DocType |
Hypergraph containers,Linear equations | Linear equation,Discrete mathematics,Combinatorics,Vertex (geometry),Hypergraph,Constraint graph,Mathematics | Journal |
Volume | Issue | ISSN |
121 | C | Journal of Combinatorial Theory, Series B 121 (2016) 248-283 |
Citations | PageRank | References |
3 | 0.49 | 15 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| David Saxton | 1 | 13 | 3.15 |
| Andrew Thomason | 2 | 71 | 16.01 |