Star-factors of tournaments

Let S_m denote the m-vertex simple digraph formed by m-1 edges with a common tail. Let f(m) denote the minimum n such that every n-vertex tournament has a spanning subgraph consisting of n/m disjoint copies of S_m. We prove that m lg m - m lg lg m <= f(m) <= 4m^2 - 6m for sufficiently large m.Comment: 5 pages, 1 figur

Large unavoidable subtournaments

Let $D_k$ denote the tournament on $3k$ vertices consisting of three disjoint vertex classes $V_1, V_2$ and $V_3$ of size $k$, each of which is oriented as a transitive subtournament, and with edges directed from $V_1$ to $V_2$, from $V_2$ to $V_3$ and from $V_3$ to $V_1$. Fox and Sudakov proved that given a natural number $k$ and $\epsilon > 0$ there is $n_0(k,\epsilon )$ such that every tournament of order $n_0(k,\epsilon )$ which is $\epsilon$-far from being transitive contains $D_k$ as a subtournament. Their proof showed that $n_0(k,\epsilon ) \leq \epsilon ^{-O(k/\epsilon ^2)}$ and they conjectured that this could be reduced to $n_0(k,\epsilon ) \leq \epsilon ^{-O(k)}$. Here we prove this conjecture.Comment: 9 page

Largest Digraphs Contained IN All N-tournaments

Let f(n) (resp. g(n)) be the largest m such that there is a digraph (resp. a spanning weakly connected digraph) on n-vertices and m edges which is a subgraph of every tournament on n-vertices. We prove that n log2 n--cxn&gt;=f(n) ~_g(n) ~- n log ~ n--c..n loglog n