arXiv subjects

Peter Allen

Publications and source records attributed to Peter Allen.

59 records · Page 4Linked to original sources

arXiv2011

Turánnical hypergraphs

This paper is motivated by the question of how global and dense restriction sets in results from extremal combinatorics can be replaced by less global and sparser ones. The result we consider here as an example is Turan's theorem, which deals with graphs G=([n],E) such that no member of the restriction set consisting of all r-tuples on [n] induces a copy of K_r. Firstly, we examine what happens when this restriction set is replaced just by all r-tuples touching a given m-element set. That is, we determine the maximal number of edges in an n-vertex such that no K_r hits a given vertex set. Secondly, we consider sparse random restriction sets. An r-uniform hypergraph R on vertex set [n] is called Turannical (respectively epsilon-Turannical), if for any graph G on [n] with more edges than the Turan number ex(n,K_r) (respectively (1+\eps)ex(n,K_r), no hyperedge of R induces a copy of K_r in G. We determine the thresholds for random r-uniform hypergraphs to be Turannical and to epsilon-Turannical. Thirdly, we transfer this result to sparse random graphs, using techniques recently developed by Schacht [Extremal results for random discrete structures] to prove the Kohayakawa-Luczak-Rodl Conjecture on Turan's theorem in random graphs.

math.CO↗

arXiv2011

Dense H-free graphs are almost (χ(H)-1)-partite

By using the Szemerédi Regularity Lemma, Alon and Sudakov recently extended the classical Andrásfai-Erdõs-Sós theorem to cover general graphs. We prove, without using the Regularity Lemma, that the following stronger statement is true. Given any (r-1)-partite graph H whose smallest part has t vertices, and any fixed c>0, there exists a constant C such that whenever G is an n-vertex graph with minimum degree at least ((3r-4)/(3r-1)+c)n, either G contains H, or we can delete at most Cn^(2-1/t) edges from G to yield an r-partite graph.

math.CO↗

arXiv2010

Filling the gap between Turán's theorem and Pósa's conjecture

Much of extremal graph theory has concentrated either on finding very small subgraphs of a large graph (Turan-type results) or on finding spanning subgraphs (Dirac-type results). In this paper we are interested in finding intermediate-sized subgraphs. We investigate minimum degree conditions under which a graph G contains squared paths and squared cycles of arbitrary specified lengths. We determine precise thresholds, assuming that the order of G is large. This extends results of Fan and Kierstead [J. Combin. Theory Ser. B 63 (1995), 55--64] and of Komlos, Sarkozy, and Szemeredi [Random Structures Algorithms 9 (1996), 193--211] concerning the containment of a spanning squared path and a spanning squared cycle, respectively. Our results show that such minimum degree conditions constitute not merely an interpolation between the corresponding Turan-type and Dirac-type results, but exhibit other interesting phenomena.

math.CO↗

arXiv2010

Ramsey-goodness -- and otherwise

A celebrated result of Chvátal, Rödl, Szemerédi and Trotter states (in slightly weakened form) that, for every natural number $Δ$, there is a constant $r_Δ$ such that, for any connected $n$-vertex graph $G$ with maximum degree $Δ$, the Ramsey number $R(G,G)$ is at most $r_Δn$, provided $n$ is sufficiently large. In 1987, Burr made a strong conjecture implying that one may take $r_Δ= Δ$. However, Graham, Rödl and Ruciński showed, by taking $G$ to be a suitable expander graph, that necessarily $r_Δ> 2^{cΔ}$ for some constant $c>0$. We show that the use of expanders is essential: if we impose the additional restriction that the bandwidth of $G$ be at most some function $β(n) = o(n)$, then $R(G,G) \le (2χ(G)+4)n\leq (2Δ+6)n$, i.e., $r_Δ= 2Δ+6$ suffices. On the other hand, we show that Burr's conjecture itself fails even for $P_n^k$, the $k$th power of a path $P_n$. Brandt showed that for any $c$, if $Δ$ is sufficiently large, there are connected $n$-vertex graphs $G$ with $Δ(G)\leqΔ$ but $R(G,K_3)>cn$. We show that, given $Δ$ and $H$, there are $β>0$ and $n_0$ such that, if $G$ is a connected graph on $n\ge n_0$ vertices with maximum degree at most $Δ$ and bandwidth at most $βn$, then we have $R(G,H)=(χ(H)-1)(n-1)+σ(H)$, where $σ(H)$ is the smallest size of any part in any $χ(H)$-partition of $H$. We also show that the same conclusion holds without any restriction on the maximum degree of $G$ if the bandwidth of $G$ is at most $ε(H) \log n/\log\log n$.

math.CO↗

arXiv2006

Not Alone: Tracing the Origins of Very Low Mass Stars and Brown Dwarfs Through Multiplicity Studies

The properties of multiple stellar systems have long provided important empirical constraints for star formation theories, enabling (along with several other lines of evidence) a concrete, qualitative picture of the birth and early evolution of normal stars. At very low masses (VLM; M <~ 0.1 M_sun), down to and below the hydrogen burning minimum mass, our understanding of formation processes is not as clear, with several competing theories now under consideration. One means of testing these theories is through the empirical characterization of VLM multiple systems. Here, we review the results of various VLM multiplicity studies to date. These systems can be generally characterized as closely separated (93% have projected separations Delta < 20 AU) and near equal-mass (77% have M_2/M_1 >= 0.8) occurring infrequently (perhaps 10-30%). Both the frequency and maximum separation of stellar and brown dwarf binaries steadily decrease for lower system masses, suggesting that VLM binary formation and/or evolution may be a mass-dependent process. There is evidence for a fairly rapid decline in the number of loosely-bound systems below ~0.3 M_sun, corresponding to a factor of 10-20 increase in the minimum binding energy of VLM binaries as compared to more massive stellar binaries. This wide-separation ``desert'' is present among both field (~1-5 Gyr) and older (> 100 Myr) cluster systems, while the youngest (<~10 Myr) VLM binaries, particularly those in nearby, low-density star forming regions, appear to have somewhat different systemic properties. We compare these empirical trends to predictions laid out by current formation theories, and outline future observational studies needed to probe the full parameter space of the lowest mass multiple systems.

astro-ph↗