Proof Wanted

Let $\mathcal{F}$ be a downward-closed (ideal) family of finite sets. Then the largest intersecting subfamily of $\mathcal{F}$ is a star: there exists an element $x$ such that the maximum size of an intersecting subfamily equals $|\{A \in \mathcal{F} : x \in A\}|$.

The Davenport constant $D(G)$ of a finite abelian group $G$ is the smallest integer $d$ such that every multiset of $d$ elements from $G$ contains a non-empty zero-sum subsequence.

Conjecture: $D(\mathbb{Z}_n \oplus \mathbb{Z}_n) = 2n - 1$.

The lower bound $D(\mathbb{Z}_n \oplus \mathbb{Z}_n) \geq 2n - 1$ is easy (take $n-1$ copies of $(1,0)$ and $n-1$ copies of $(0,1)$). The upper bound is the content of the conjecture.

The equational_theories project catalogues thousands of equational laws for magmas (types with a single binary operation) and maps their logical relationships. Many implications remain undecided: given two equations, it is unknown whether every magma satisfying the first must also satisfy the second.

Sample equations:

• Equation 46: $x \cdot (y \cdot x) = y$
• Equation 387: $x \cdot y = (y \cdot x) \cdot x$
• Equation 4512: $x \cdot (y \cdot z) = (x \cdot y) \cdot z$ (associativity)

The maximum size of a $k$-uniform family on $[n]$ with no $s+1$ pairwise disjoint sets is

$\max\left\{\binom{ks+k-1}{k}, \binom{n}{k} - \binom{n-s}{k}\right\}$

For every integer $n \geq 2$, the equation

$\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$

has a solution in positive integers $x, y, z$.

Frankl's union-closed conjecture (lattice-theoretic form): In every finite lattice, there exists a join-irreducible element $j$ such that $j$ is above at most half the elements:

$2 \cdot |\{x \in L : j \leq x\}| \leq |L|$

We state the conjecture restricted to upper semimodular lattices: lattices satisfying $a \wedge b \lessdot a \implies b \lessdot a \vee b$.

For every Boolean function $f : \{0,1\}^n \to \{0,1\}$,

$\mathrm{bs}(f) \leq \mathrm{s}(f)^2$

where $\mathrm{s}(f)$ is the sensitivity and $\mathrm{bs}(f)$ is the block sensitivity.

There exists a constant $C$ such that any family of more than $C^k$ sets, each of size $k$, contains a sunflower of size 3 (i.e., three sets whose pairwise intersections are all equal).