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

Embedding dimensions of simplicial complexes on few vertices

We provide a simple characterization of simplicial complexes on few vertices that embed into the $d$-sphere. Namely, a simplicial complex on $d+3$ vertices embeds into the $d$-sphere if and only if its non-faces do not form an intersecting family. As immediate consequences, we recover the classical van Kampen--Flores theorem and provide a topological extension of the Erd\H os--Ko--Rado theorem. By analogy with F\'ary's theorem for planar graphs, we show in addition that such complexes satisfy the rigidity property that continuous and linear embeddability are equivalent.Comment: 8 page

High dimensional Hoffman bound and applications in extremal combinatorics

One powerful method for upper-bounding the largest independent set in a graph is the Hoffman bound, which gives an upper bound on the largest independent set of a graph in terms of its eigenvalues. It is easily seen that the Hoffman bound is sharp on the tensor power of a graph whenever it is sharp for the original graph. In this paper, we introduce the related problem of upper-bounding independent sets in tensor powers of hypergraphs. We show that many of the prominent open problems in extremal combinatorics, such as the Tur\'an problem for (hyper-)graphs, can be encoded as special cases of this problem. We also give a new generalization of the Hoffman bound for hypergraphs which is sharp for the tensor power of a hypergraph whenever it is sharp for the original hypergraph. As an application of our Hoffman bound, we make progress on the problem of Frankl on families of sets without extended triangles from 1990. We show that if $\frac{1}{2}n\le2k\le\frac{2}{3}n,$ then the extremal family is the star, i.e. the family of all sets that contains a given element. This covers the entire range in which the star is extremal. As another application, we provide spectral proofs for Mantel's theorem on triangle-free graphs and for Frankl-Tokushige theorem on $k$-wise intersecting families

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

Cross-intersecting non-empty uniform subfamilies of hereditary families

Two families A and B of sets are cross-t-intersecting if each set in A intersects each set in B in at least t elements. A family H is hereditary if for each set A in H, all the subsets of A are in H. Let H(r) denote the family of r-element sets in H. We show that for any integers t, r, and s with 1 ≤ t ≤ r ≤ s, there exists an integer c(r, s, t) such that the following holds for any hereditary family H whose maximal sets are of size at least c(r, s, t). If A is a nonempty subfamily of H(r) , B is a non-empty subfamily of H(s) , A and B are cross-t-intersecting, and |A| + |B| is maximum under the given conditions, then for some set I in H with t ≤ |I| ≤ r, either A = {A ∈ H(r) : I ⊆ A} and B = {B ∈ H(s) : |B ∩ I| ≥ t}, or r = s, t < |I|, A = {A ∈ H(r) : |A ∩ I| ≥ t}, and B = {B ∈ H(s) : I ⊆ B}. We give c(r, s, t) explicitly. The result was conjectured by the author for t = 1 and generalizes well-known results for the case where H is a power set.peer-reviewe