Rational exponents in extremal graph theory

Given a family of graphs $\mathcal{H}$, the extremal number $\textrm{ex}(n, \mathcal{H})$ is the largest $m$ for which there exists a graph with $n$ vertices and $m$ edges containing no graph from the family $\mathcal{H}$ as a subgraph. We show that for every rational number $r$ between $1$ and $2$, there is a family of graphs $\mathcal{H}_r$ such that $\textrm{ex}(n, \mathcal{H}_r) = \Theta(n^r)$. This solves a longstanding problem in the area of extremal graph theory.Comment: 11 pages. arXiv admin note: text overlap with arXiv:1411.085

Applications of entropy to extremal problems

The Sidorenko conjecture gives a lower bound on the number of homomorphisms from a bipartite graph to another graph. Szegedy [28] used entropy methods to prove the conjecture in some cases. We will refine these methods to also give lower bounds for the number of injective homomorphisms from a bipartite graph to another bipartite graph, and a lower bound for the number of homomorphisms from a k-partite hypergraph to another k-partite hypergraph, as well as a few other similar problems. Next is a generalisation of the Kruskal Katona Theorem [19, 17]. We are given integers k 4 we will make a lot of progress towards finding a solution. The next chapter is to do with Turán-type problems. Given a family of k-hypergraphs F, ex(n;F) is the maximum number of edges an F-free n-vertex k-hypergraph can have. We prove that for a rational r, there exists some finite family F of k-hypergraphs for which ex(n;F) = Ɵ(nk-r) if and only if 0 < r < k - 1 or r = k. The final chapter will deal with the implicit representation conjecture, in the special case of semi-algebraic graphs. Given a graph in such a family, we want to assign a name to each vertex in such a way that we can recover each edge based only on the names of the two incident vertices. We will first prove that one `obvious' way of storing the information doesn't work. Then we will come up with a way of storing the information that requires O(n1-E) bits per vertex, where E is some small constant depending only on the family

Invitation to intersection problems for finite sets

Extremal set theory is dealing with families, . F of subsets of an . n-element set. The usual problem is to determine or estimate the maximum possible size of . F, supposing that . F satisfies certain constraints. To limit the scope of this survey most of the constraints considered are of the following type: any . r subsets in . F have at least . t elements in common, all the sizes of pairwise intersections belong to a fixed set, . L of natural numbers, there are no . s pairwise disjoint subsets. Although many of these problems have a long history, their complete solutions remain elusive and pose a challenge to the interested reader.Most of the paper is devoted to sets, however certain extensions to other structures, in particular to vector spaces, integer sequences and permutations are mentioned as well. The last part of the paper gives a short glimpse of one of the very recent developments, the use of semidefinite programming to provide good upper bound