On the classification of Stanley sequences

An integer sequence is said to be 3-free if no three elements form an arithmetic progression. Following the greedy algorithm, the Stanley sequence $S(a_0,a_1,\ldots,a_k)$ is defined to be the 3-free sequence $\{a_n\}$ having initial terms $a_0,a_1,\ldots,a_k$ and with each subsequent term $a_n>a_{n-1}$ chosen minimally such that the 3-free condition is not violated. Odlyzko and Stanley conjectured that Stanley sequences divide into two classes based on asymptotic growth patterns, with one class of highly structured sequences satisfying $a_n\approx \Theta(n^{\log_2 3})$ and another class of seemingly chaotic sequences obeying $a_n=\Theta(n^2/\log n)$. We propose a rigorous definition of regularity in Stanley sequences based on local structure rather than asymptotic behavior and show that our definition implies the corresponding asymptotic property proposed by Odlyzko and Stanley. We then construct many classes of regular Stanley sequences, which include as special cases all such sequences previously identified. We show how two regular sequences may be combined into another regular sequence, and how parts of a Stanley sequence may be translated while preserving regularity. Finally, we demonstrate that certain Stanley sequences possess proper subsets that are also Stanley sequences, a situation that appears previously to have been assumed impossible.Comment: 25 page

Acyclic Subgraphs of Planar Digraphs

An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on $n$ vertices without directed 2-cycles possesses an acyclic set of size at least $3n/5$. We prove this conjecture for digraphs where every directed cycle has length at least 8. More generally, if $g$ is the length of the shortest directed cycle, we show that there exists an acyclic set of size at least $(1 - 3/g)n$.Comment: 9 page

Trees with an On-Line Degree Ramsey Number of Four

On-line Ramsey theory studies a graph-building game between two players. The player called Builder builds edges one at a time, and the player called Painter paints each new edge red or blue after it is built. The graph constructed is called the background graph. Builder's goal is to cause the background graph to contain a monochromatic copy of a given goal graph, and Painter's goal is to prevent this. In the S[subscript k]-game variant of the typical game, the background graph is constrained to have maximum degree no greater than k. The on-line degree Ramsey number [˚over R][subscript Δ](G) of a graph G is the minimum k such that Builder wins an S[subscript k]-game in which G is the goal graph. Butterfield et al. previously determined all graphs G satisfying [˚ over R][subscript Δ](G)≤3. We provide a complete classification of trees T satisfying [˚ over R][subscript Δ](T)=4.National Science Foundation (U.S.) (Grant DMS-0754106)United States. National Security Agency (Grant H98230-06-1-0013