arXiv subjects
Publications and source records attributed to Luka Milićević.
Let $f_r(n)$ be the minimum number of complete $r$-partite $r$-graphs needed to partition the edge set of the complete $r$-uniform hypergraph on $n$ vertices. Graham and Pollak showed that $f_2(n) = n-1$. An easy construction shows that $f_r(n)\le (1-o(1))\binom{n}{\lfloor r/2\rfloor}$ and it has been unknown if this upper bound is asymptotically sharp. In this paper we show that $f_r(n)\le (\frac{14}{15}+o(1))\binom{n}{r/2}$ for each even $r\ge 4$.
Erdős asked the following question: given $n$ points in the plane in almost general position (no 4 collinear), how large a set can we guarantee to find that is in general position (no 3 collinear)? Füredi constructed a set of $n$ points in almost general position with no more than $o(n)$ points in general position. Cardinal, Tóth and Wood extended this result to $\mathbb{R}^3$, finding sets of $n$ points with no 5 on a plane whose subsets with no 4 points on a plane have size $o(n)$, and asked the question for higher dimensions: for given $n$, is it still true that the largest subset in general position we can guarantee to find has size $o(n)$? We answer their question for all $d$ and derive improved bounds for certain dimensions.
A family $f_1,...,f_n$ of operators on a complete metric space $X$ is called contractive if there exists a positive $λ< 1$ such that for any $x,y$ in $X$ we have $d(f_i(x),f_i(y)) \leq λd(x,y)$ for some $i$. Austin conjectured that any commuting contractive family of operators has a common fixed point, and he proved this for the case of two operators. Our aim in this paper is to show that Austin's conjecture is true for three operators, provided that $λ$ is sufficiently small.