Given a graph $G=(V,E)$ and a colouring $f:Emapsto mathbb N$, the induced colour of a vertex $v$ is the sum of the colours at the edges incident with $v$. If all the induced colours of vertices of $G$ are distinct, the colouring is called antimagic.
If $G$ has a bijective antimagic colouring $f:Emapsto {1,dots,|E|}$, the graph $G$ is called antimagic. A conjecture of Hartsfield and Ringel states that all connected graphs other than $K_2$ are antimagic. Alon, Kaplan, Lev, Roddity and Yuster proved this conjecture for graphs with minimum degree at least $c log |V|$ for some constant $c$; we improve on this result, proving the conjecture for graphs with average degree at least some constant $d_0$.
A family of sets is called union-closed if whenever $A$ and $B$ are sets of the family, so is $Acup B$. The long-standing union-closed conjecture states that if a family of subsets of $[n]$ is union-closed, some element appears in at least half the s
ets of the family. A natural weakening is that the union-closed conjecture holds for large families; that is, families consisting of at least $p_02^n$ sets for some constant $p_0$. The first result in this direction appears in a recent paper of Balla, Bollobas and Eccles cite{BaBoEc}, who showed that union-closed families of at least $frac{2}{3}2^n$ sets satisfy the conjecture --- they proved this by determining the minimum possible average size of a set in a union-closed family of given size. However, the methods used in that paper cannot prove a better constant than $frac{2}{3}$. Here, we provide a stability result for the main theorem of cite{BaBoEc}, and as a consequence we prove the union-closed conjecture for families of at least $(frac{2}{3}-c)2^n$ sets, for a positive constant $c$.
For integers $m_1,...,m_d>0$ and a cuboid $M=[0,m_1]times ... times [0,m_d]subset mathbb{R}^d$, a brick of $M$ is a closed cuboid whose vertices have integer coordinates. A set $H$ of bricks in $M$ is a system of brick islands if for each pair of bri
cks in $H$ one contains the other or they are disjoint. Such a system is maximal if it cannot be extended to a larger system of brick islands. Extending the work of Lengv{a}rszky, we show that the minimum size of a maximal system of brick islands in $M$ is $sum_{i=1}^d m_i - (d-1)$. Also, in a cube $C=[0,m]^d$ we define the corresponding notion of a system of cubic islands, and prove bounds on the sizes of maximal systems of cubic islands.
