Strongly intersecting integer partitions

We call a sum a1+a2+• • •+ak a partition of n of length k if a1, a2, . . . , ak and n are positive integers such that a1 ≤ a2 ≤ • • • ≤ ak and n = a1 + a2 + • • • + ak. For i = 1, 2, . . . , k, we call ai the ith part of the sum a1 + a2 + • • • + ak. Let Pn,k be the set of all partitions of n of length k. We say that two partitions a1+a2+• • •+ak and b1+b2+• • •+bk strongly intersect if ai = bi for some i. We call a subset A of Pn,k strongly intersecting if every two partitions in A strongly intersect. Let Pn,k(1) be the set of all partitions in Pn,k whose first part is 1. We prove that if 2 ≤ k ≤ n, then Pn,k(1) is a largest strongly intersecting subset of Pn,k, and uniquely so if and only if k ≥ 4 or k = 3 ≤ n ̸∈ {6, 7, 8} or k = 2 ≤ n ≤ 3.peer-reviewe

Cross-intersecting non-empty uniform subfamilies of hereditary families

A set $A$ $t$-intersects a set $B$ if $A$ and $B$ have at least $t$ common elements. A set of sets is called a family. Two families $\mathcal{A}$ and $\mathcal{B}$ are cross-$t$-intersecting if each set in $\mathcal{A}$ $t$-intersects each set in $\mathcal{B}$. A family $\mathcal{H}$ is hereditary if for each set $A$ in $\mathcal{H}$, all the subsets of $A$ are in $\mathcal{H}$. The $r$th level of $\mathcal{H}$, denoted by $\mathcal{H}^{(r)}$, is the family of $r$-element sets in $\mathcal{H}$. A set $B$ in $\mathcal{H}$ is a base of $\mathcal{H}$ if for each set $A$ in $\mathcal{H}$, $B$ is not a proper subset of $A$. Let $\mu(\mathcal{H})$ denote the size of a smallest base of $\mathcal{H}$. We show that for any integers $t$, $r$, and $s$ with $1 \leq t \leq r \leq s$, there exists an integer $c(r,s,t)$ such that the following holds for any hereditary family $\mathcal{H}$ with $\mu(\mathcal{H}) \geq c(r,s,t)$. If $\mathcal{A}$ is a non-empty subfamily of $\mathcal{H}^{(r)}$, $\mathcal{B}$ is a non-empty subfamily of $\mathcal{H}^{(s)}$, $\mathcal{A}$ and $\mathcal{B}$ are cross-$t$-intersecting, and $|\mathcal{A}| + |\mathcal{B}|$ is maximum under the given conditions, then for some set $I$ in $\mathcal{H}$ with $t \leq |I| \leq r$, either $\mathcal{A} = \{A \in \mathcal{H}^{(r)} \colon I \subseteq A\}$ and $\mathcal{B} = \{B \in \mathcal{H}^{(s)} \colon |B \cap I| \geq t\}$, or $r = s$, $t < |I|$, $\mathcal{A} = \{A \in \mathcal{H}^{(r)} \colon |A \cap I| \geq t\}$, and $\mathcal{B} = \{B \in \mathcal{H}^{(s)} \colon I \subseteq B\}$. This was conjectured by the author for $t=1$ and generalizes well-known results for the case where $\mathcal{H}$ is a power set.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1805.0524

Non-trivial intersecting uniform sub-families of hereditary families

For a family F of sets, let μ(F ) denote the size of a smallest set in F that is not a subset of any other set in F , and for any positive integer r, let F (r) denote the family of r-element sets in F . We say that a family A is of Hilton–Milner (HM) type if for some A ∈ A, all sets in A \ {A} have a common element x ̸∈ A and intersect A. We show that if a hereditary family H is compressed and μ(H) ≥ 2r ≥ 4, then the HM-type family {A ∈ H(r): 1 ∈ A, A∩[2,r+1] ̸= ∅}∪{[2,r+1]}is a largest non-trivial intersecting sub-family of H(r); this generalises a well-known result of Hilton and Milner. We demonstrate that for any r ≥ 3 and m ≥ 2r, there exist non-compressed hereditary families H with μ(H) = m such that no largest non-trivial intersecting sub-family of H(r) is of HM type, and we suggest two conjectures about the extremal structures for arbitrary hereditary families.peer-reviewe

Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs

In this thesis we investigate three different aspects of graph theory. Firstly, we consider interesecting systems of independent sets in graphs, and the extension of the classical theorem of Erdos, Ko and Rado to graphs. Our main results are a proof of an Erdos-Ko-Rado type theorem for a class of trees, and a class of trees which form counterexamples to a conjecture of Hurlberg and Kamat, in such a way that extends the previous counterexamples given by Baber. Secondly, we investigate perfect graphs - specifically, edge modification aspects of perfect graphs and their subclasses. We give some alternative characterisations of perfect graphs in terms of edge modification, as well as considering the possible connection of the critically perfect graphs - previously studied by Wagler - to the Strong Perfect Graph Theorem. We prove that the situation where critically perfect graphs arise has no analogue in seven different subclasses of perfect graphs (e.g. chordal, comparability graphs), and consider the connectivity of a bipartite reconfiguration-type graph associated to each of these subclasses. Thirdly, we consider a graph theoretic structure called a bireflexive graph where every vertex is both adjacent and nonadjacent to itself, and use this to characterise modular decompositions as the surjective homomorphisms of these structures. We examine some analogues of some graph theoretic notions and define a “dual” version of the reconstruction conjecture

On multicolor Ramsey numbers of triple system paths of length 3

Let $\mathcal{H}$ be a 3-uniform hypergraph. The multicolor Ramsey number $r_k(\mathcal{H})$ is the smallest integer $n$ such that every coloring of $\binom{[n]}{3}$ with $k$ colors has a monochromatic copy of $\mathcal{H}$. Let $\mathcal{L}$ be the loose 3-uniform path with 3 edges and $\mathcal{M}$ denote the messy 3-uniform path with 3 edges; that is, let $\mathcal{L} = \{abc, cde, efg\}$ and $\mathcal{M} = \{ abc, bcd, def\}$. In this note we prove $r_k(\mathcal{L}) < 1.55k$ and $r_k(\mathcal{M}) < 1.6k$ for $k$ sufficiently large. The former result improves on the bound $r_k( \mathcal{L}) < 1.975k + 7\sqrt{k}$, which was recently established by {\L}uczak and Polcyn.Comment: 18 pages, 3 figure