54,088 research outputs found
Subgraph densities in signed graphons and the local Sidorenko conjecture
We prove inequalities between the densities of various bipartite subgraphs in
signed graphs and graphons. One of the main inequalities is that the density of
any bipartite graph with girth r cannot exceed the density of the r-cycle. This
study is motivated by Sidorenko's conjecture, which states that the density of
a bipartite graph F with m edges in any graph G is at least the m-th power of
the edge density of G. Another way of stating this is that the graph G with
given edge density minimizing the number of copies of F is, asymptotically, a
random graph. We prove that this is true locally, i.e., for graphs G that are
"close" to a random graph.Comment: 20 page
Product decompositions of quasirandom groups and a Jordan type theorem
We first note that a result of Gowers on product-free sets in groups has an
unexpected consequence: If k is the minimal degree of a representation of the
finite group G, then for every subset B of G with we have
B^3 = G.
We use this to obtain improved versions of recent deep theorems of Helfgott
and of Shalev concerning product decompositions of finite simple groups, with
much simpler proofs.
On the other hand, we prove a version of Jordan's theorem which implies that
if k>1, then G has a proper subgroup of index at most ck^2 for some absolute
constant c, hence a product-free subset of size at least . This
answers a question of Gowers.Comment: 18 pages. In this third version we added an Appendix with a short
proof of Proposition
Small doubling in groups
Let A be a subset of a group G = (G,.). We will survey the theory of sets A
with the property that |A.A| <= K|A|, where A.A = {a_1 a_2 : a_1, a_2 in A}.
The case G = (Z,+) is the famous Freiman--Ruzsa theorem.Comment: 23 pages, survey article submitted to Proceedings of the Erdos
Centenary conferenc
Normalizers of Primitive Permutation Groups
Let be a transitive normal subgroup of a permutation group of finite
degree . The factor group can be considered as a certain Galois group
and one would like to bound its size. One of the results of the paper is that
if is primitive unless , , , , or
. This bound is sharp when is prime. In fact, when is
primitive, unless is a member of a given infinite
sequence of primitive groups and is different from the previously listed
integers. Many other results of this flavor are established not only for
permutation groups but also for linear groups and Galois groups.Comment: 44 pages, grant numbers updated, referee's comments include
OV Graphs Are (Probably) Hard Instances
© Josh Alman and Virginia Vassilevska Williams. A graph G on n nodes is an Orthogonal Vectors (OV) graph of dimension d if there are vectors v1, . . ., vn ∈ {0, 1}d such that nodes i and j are adjacent in G if and only if hvi, vji = 0 over Z. In this paper, we study a number of basic graph algorithm problems, except where one is given as input the vectors defining an OV graph instead of a general graph. We show that for each of the following problems, an algorithm solving it faster on such OV graphs G of dimension only d = O(log n) than in the general case would refute a plausible conjecture about the time required to solve sparse MAX-k-SAT instances: Determining whether G contains a triangle. More generally, determining whether G contains a directed k-cycle for any k ≥ 3. Computing the square of the adjacency matrix of G over Z or F2. Maintaining the shortest distance between two fixed nodes of G, or whether G has a perfect matching, when G is a dynamically updating OV graph. We also prove some complementary results about OV graphs. We show that any problem which is NP-hard on constant-degree graphs is also NP-hard on OV graphs of dimension O(log n), and we give two problems which can be solved faster on OV graphs than in general: Maximum Clique, and Online Matrix-Vector Multiplication
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