A Generalization of Erdős' Matching Conjecture

Let $\mathcal{H}=(V,\mathcal{E})$ be an $r$-uniform hypergraph on $n$ vertices and fix a positive integer $k$ such that $1\le k\le r$. A $k$-matching of $\mathcal{H}$ is a collection of edges $\mathcal{M}\subset \mathcal{E}$ such that every subset of $V$ whose cardinality equals $k$ is contained in at most one element of $\mathcal{M}$. The $k$-matching number of $\mathcal{H}$ is the maximum cardinality of a $k$-matching. A well-known problem, posed by Erdős, asks for the maximum number of edges in an $r$-uniform hypergraph under constraints on its $1$-matching number. In this article we investigate the more general problem of determining the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices subject to the constraint that its $k$-matching number is strictly less than $a$. The problem can also be seen as a generalization of the well-known $k$-intersection problem. We propose candidate hypergraphs for the solution of this problem, and show that the extremal hypergraph is among this candidate set when $n\ge 4r\binom{r}{k}^2\cdot a$.

Extremal Hypergraphs for the Biased Erdős-Selfridge Theorem

A positional game is essentially a generalization of Tic-Tac-Toe played on a hypergraph $(V,{\cal F}).$ A pivotal result in the study of positional games is the Erdős–Selfridge theorem, which gives a simple criterion for the existence of a Breaker's winning strategy on a finite hypergraph ${\cal F}$. It has been shown that the bound in the Erdős–Selfridge theorem can be tight and that numerous extremal hypergraphs exist that demonstrate the tightness of the bound. We focus on a generalization of the Erdős–Selfridge theorem proven by Beck for biased $(p:q)$ games, which we call the $(p:q)$–Erdős–Selfridge theorem. We show that for $pn$-uniform hypergraphs there is a unique extremal hypergraph for the $(p:q)$–Erdős–Selfridge theorem when $q\geq 2$.

Forbidden Submatrices: Some New Bounds and Constructions

We explore an extremal hypergraph problem for which both the vertices and edges are ordered. Given a hypergraph $F$ (not necessarily simple), we consider how many edges a simple hypergraph (no repeated edges) on $m$ vertices can have while forbidding $F$ as a subhypergraph where both hypergraphs have fixed vertex and edge orderings. A hypergraph of $n$ edges on $m$ vertices can be encoded as an $m\times n$ (0,1)-matrix. We say a matrix is simple if it is a (0,1)-matrix with no repeated columns. Given a (0,1)-matrix $F$, we define ${\hbox{fs}}(m,F)$ as the maximum, over all simple matrices $A$ which do not have $F$ as a submatrix, of the number of columns in $A$. The row and column order matter. It is known that if $F$ is $k\times \ell$ then ${\hbox{fs}}(m,F)$ is $O(m^{2k-1-\epsilon})$ where $\epsilon=(k-1)/(13\log_2 \ell)$. Anstee, Frankl, Füredi and Pach have conjectured that if $F$ is $k$-rowed, then ${\hbox{fs}}(m,F)$ is $O(m^k)$. We show ${\hbox{fs}}(m,F)$ is $O(m^2)$ for $F= \left[{1\,0\,1\,0\,1\atop 0\,1\,0\,1\,0}\cdots\right]$ and for $F= \left[{1\,0\,1\,0\,1\atop 1\,0\,1\,0\,1}\cdots\right]$. The proofs use a type of amortized analysis. We also give some constructions.