We show that a number of conditions on oriented graphs, all of which are satisfied with high probability by randomly oriented graphs, are equivalent. These equivalences are similar to those given by Chung, Graham and Wilson in the case of unoriented
graphs, and by Chung and Graham in the case of tournaments. Indeed, our main theorem extends to the case of a general underlying graph G the main result of Chung and Graham which corresponds to the case that G is complete. One interesting aspect of these results is that exactly two of the four orientations of a four-cycle can be used for a quasi-randomness condition, i.e., if the number of appearances they make in D is close to the expected number in a random orientation of the same underlying graph, then the same is true for every small oriented graph H
This paper has been withdrawn by the author.
A $k$-sum of a set $Asubseteq mathbb{Z}$ is an integer that may be expressed as a sum of $k$ distinct elements of $A$. How large can the ratio of the number of $(k+1)$-sums to the number of $k$-sums be? Writing $kwedge A$ for the set of $k$-sums of $
A$ we prove that [ frac{|(k+1)wedge A|}{|kwedge A|}, le , frac{|A|-k}{k+1} ] whenever $|A|ge (k^{2}+7k)/2$. The inequality is tight -- the above ratio being attained when $A$ is a geometric progression. This answers a question of Ruzsa.
It is an intriguing question to see what kind of information on the structure of an oriented graph $D$ one can obtain if $D$ does not contain a fixed oriented graph $H$ as a subgraph. The related question in the unoriented case has been an active are
a of research, and is relatively well-understood in the theory of quasi-random graphs and extremal combinatorics. In this paper, we consider the simplest cases of such a general question for oriented graphs, and provide some results on the global behavior of the orientation of $D$. For the case that $H$ is an oriented four-cycle we prove: in every $H$-free oriented graph $D$, there is a pair $A,Bssq V(D)$ such that $e(A,B)ge e(D)^{2}/32|D|^{2}$ and $e(B,A)le e(A,B)/2$. We give a random construction which shows that this bound on $e(A,B)$ is best possible (up to the constant). In addition, we prove a similar result for the case $H$ is an oriented six-cycle, and a more precise result in the case $D$ is dense and $H$ is arbitrary. We also consider the related extremal question in which no condition is put on the oriented graph $D$, and provide an answer that is best possible up to a multiplicative constant. Finally, we raise a number of related questions and conjectures.
For a subset A of a finite abelian group G we define Sigma(A)={sum_{ain B}a:Bsubset A}. In the case that Sigma(A) has trivial stabiliser, one may deduce that the size of Sigma(A) is at least quadratic in |A|; the bound |Sigma(A)|>= |A|^{2}/64 has rec
ently been obtained by De Vos, Goddyn, Mohar and Samal. We improve this bound to the asymptotically best possible result |Sigma(A)|>= (1/4-o(1))|A|^{2}. We also study a related problem in which A is any subset of Z_{n} with all elements of A coprime to n; it has recently been shown, by Vu, that if such a set A has the property Sigma(A) is not Z_{n} then |A|=O(sqrt{n}). This bound was improved to |A|<= 8sqrt{n} by De Vos, Goddyn, Mohar and Samal, we further improve the bound to the asymptotically best possible result |A|<= (2+o(1))sqrt{n}.
