44,744 research outputs found
Multiple cross-intersecting families of signed sets
A k-signed r-set on[n] = {1, ..., n} is an ordered pair (A, f), where A is an r-subset of [n] and f is a function from A to [k]. Families A1, ..., Ap are said to be cross-intersecting if any set in any family Ai intersects any set in any other family Aj. Hilton proved a sharp bound for the sum of sizes of cross-intersecting families of r-subsets of [n]. Our aim is to generalise Hilton's bound to one for families of k-signed r-sets on [n]. The main tool developed is an extension of Katona's cyclic permutation argument.peer-reviewe
A Hilton–Milner-type theorem and an intersection conjecture for signed sets
A family A of sets is said to be intersecting if any two sets in A intersect (i.e. have at least one common element). A is said to be centred if there is an element common to all the sets in A; otherwise, A is said to be non-centred. For any r ∈ [n] := {1, . . . , n} and any integer k ≥ 2, let Sn,r,k be the family {{(x1, y1), . . . , (xr, yr)}: x1, . . . , xr are distinct elements of [n], y1, . . . , yr ∈ [k]} of k-signed r-sets on [n]. Let m := max{0, 2r−n}.We establish the following Hilton–Milner-type theorems, the second of which is proved using the first: (i) If A1 and A2 are non-empty cross-intersecting (i.e. any set in A1 intersects any set in A2) sub-families of Sn,r,k, then |A1| + |A2| ≤ n R K r −r i=m r I (k − 1) I n – r r – I K r−i + 1. (ii) If A is a non-centred intersecting sub-family of Sn,r,k, 2 ≤ r ≤ n, then |A| ≤ n – 1 r – 1 K r−1 −r−1 i=m r I (k − 1) I n − 1 – r r − 1 – I K r−1−i + 1 if r < n; k r−1 − (k − 1) r−1 + k − 1 if r = n. We also determine the extremal structures. (ii) is a stability theorem that extends Erdős–Ko–Rado-type results proved by various authors. We then show that (ii) leads to further evidence for an intersection conjecture suggested by the author about general signed set systems.peer-reviewe
Some results in extremal combinatorics
In Chapter 1 we determine the minimal density of triangles in a tripartite graph with prescribed edge densities. This extends work of Bondy, Shen, Thomassé and Thomassen characterizing those edge densities guaranteeing the existence of a triangle in a tripartite graph. We also determine those edge densities guaranteeing a copy of a triangle or C5 in a tripartite graph.
In Chapter 2 we describe Razborov's flag algebra method and apply this to Erdös' jumping hypergraph problem to find the first non-trivial regions of jumps. We also use Razborov's method to prove five new sharp Turan densities,
by looking at six vertex 3-graphs which are edge minimal and not 2-colourable.
We extend Razborov's method to hypercubes. This allows us to significantly improve the upper bound given by Thomason and Wagner on the number of edges in a C4-free subgraph of the hypercube. We also show that the vertex
Turan density of a 3-cube with a single vertex removed is precisely 3/4.
In Chapter 3 we look at problems for intersecting families of sets on graphs.
We give a new bound for the size of G-intersecting families on a cycle, disproving a conjecture of Johnson and Talbot. We also extend this result to cross-intersecting families and to powers of cycles.
Finally in Chapter 4 we disprove a conjecture of Hurlbert and Kamat that
the largest trivial intersecting family of independent r-sets from the vertex set
of a tree is centred on a leaf
Cross-intersecting families of vectors
Given a sequence of positive integers , let
denote the family of all sequences of positive integers
such that for all . Two families of sequences (or vectors),
, are said to be -cross-intersecting if no matter how we
select and , there are at least distinct indices
such that . We determine the maximum value of over all
pairs of - cross-intersecting families and characterize the extremal pairs
for , provided that . The case is
quite different. For this case, we have a conjecture, which we can verify under
additional assumptions. Our results generalize and strengthen several previous
results by Berge, Frankl, F\"uredi, Livingston, Moon, and Tokushige, and
answers a question of Zhang
- …