On 2-ranks of Steiner triple systems

Some derived steiner triple systems

Triple systems and binary operations

Zero-sum flows for triple systems

About 80 eclipsing binaries are known to be components of multiple star systems. This is less than 2% of all known eclipsing binary systems, but the actual total is certainly much high than this, as the majority of stars under review are in fairly bright, well studied systems. Triple star systems are by far the most common of these, but one system in five is a quadruple or higher order system. Quadruple systems include both systems of hierarchy 2 (two widely spaced pairs, e.g., BV + BW Dra) and hierarchy 3 (e.g., Eta Ori). V772 Her and DN UMa are in quintuple systems, and HR 3337 and the system that contains GZ And may also be quintuple. Castor, which contains the eclipsing binary YY Gem, is a very rare example of a sextuple star system. Also considered are the eclipsing binaries in compact star clusters such as SZ Cam and the two binaries in the Trapezium, BM Ori and V1016 Ori. The most frequently encountered type of eclipsing binary to be found in multiple star systems is that composed of two early-type (B5 or earlier) stars. Nearly all of the very close triple systems which have eclipsing binary components are of this type. W UMa-type stars are also well represented among the multiple star systems containing eclipsing binaries, but some other types are not. The RS CVn stars occur less frequently as components of multiple star systems, and cataclysmic binaries appear to occur only very rarely as components of these systems. The physical and orbital properties of the multiple star systems are discussed in some detail in cases where the data are fairly complete, but it must be noted that many of the visual pairs have periods of revolution which are much too long to permit reliable determination of orbits. In triple star systems with eclipsing binaries the ratio of P2/Pt ranges from less than 10 (for lambda Tau) to more than 107 in some cases. The questions of co-planarity of orbits and the membership of components in multiple star systems are also considered.

The binary code spanned by the rows of the point by block incidence matrix of a Steiner triple system STS(v) is studied. A sufficient condition for such a code to contain a unique equivalence class of STS(v)'s of maximal rank within the code is proved. The code of the classical Steiner triple system defined by the lines in PG(n-1,2) ( n\geq 3), or AG(n,3) ( n\geq 3) is shown to contain exactly v codewords of weight r = (v-1)/2, hence the system is characterized by its code. In addition, the code of the projective STS(2^{n}-1) is characterized as the unique (up to equivalence) binary linear code with the given parameters and weight distribution. In general, the number of STS(v)'s contained in the code depends on the geometry of the codewords of weight r. It is demonstrated that the ovals and hyperovals of the definingSTS(v) play a crucial role in this geometry. This relation is utilized for the construction of some infinite classes of Steiner triple systems without ovals.

We present a catalog (status 2022 July 1) of triple and higher-order systems identified containing exoplanets based on data from the literature, including various analyses. We explore statistical properties of the systems with a focus on both the stars and the planets. So far, about 30 triple systems and one to three quadruple systems, including (mildly) controversial cases, have been found. The total number of planets is close to 40. All planet-hosting triple-star systems are highly hierarchic, consisting of a quasi-binary complemented by a distant stellar component, which is in orbit about the common center of mass. Furthermore, the quadruple systems are in fact pairs of close binaries (“double–doubles”), with one binary harboring a planet. For the different types of star–planet systems, we introduce a template for the classifications of planetary orbital configurations in correspondence to the hierarchy of the system and the planetary host. The data show that almost all stars are main-sequence stars, as expected. However, the stellar primaries tend to be more massive (i.e., corresponding to spectral types A, F, and G) than expected from single-star statistics, a finding also valid for stellar secondaries but less pronounced. Tertiary stellar components are almost exclusively low-mass stars of spectral type M. Almost all planets have been discovered based on either the Radial Velocity method or the Transit method. Both gas giants (the dominant type) and terrestrial planets (including super-Earths) have been identified. We anticipate the expansion of this database in the light of future planetary search missions.

It is well known that when or , there exists a Steiner triple system (STS) of order n decomposable into triangles (three pairwise intersecting triples whose intersection is empty). A triangle in an STS determines naturally two more triples: the triple of “vertices” , and the triple of “midpoints” . The number of these triples in both cases, that of “vertex” triples (inner) or that of “midpoint triples” (outer), equals one-third of the number of triples in the STS. In this paper, we consider a new problem of trinal decompositions of an STS into triangles. In this problem, one asks for three distinct decompositions of an STS of order n into triangles such that the union of the three collections of inner triples (outer triples, respectively) from the three decompositions form the set of triples of an STS of the same order. These decompositions are called trinal inner and trinal outer decompositions, respectively. We settle the existence question for trinal inner decompositions completely, and for trinal outer decompositions with two possible exceptions.

We show that there are exactly 2624 isomorphism classes of Steiner triple systems on 27 points having 3-rank 24, all of which are actually resolvable. More generally, all Steiner triple systems on $$3^n$$ points having 3-rank at most $$3^n-n$$ are resolvable. Combining this observation with the lower bound on the number of such $${\mathrm {STS}}(3^n)$$ recently established by two of the present authors, we obtain a strong lower bound on the number of Kirkman triple systems on $$3^n$$ points. For instance, there are more than $$10^{99}$$ isomorphism classes of $${\mathrm {KTS}}(81)$$ .

ABSTRACTA Latin square of order is an matrix in which each row and column contains each of symbols exactly once. For , we show that with high probability a uniformly random Latin square of order has no proper subsquare of order larger than . Using this fact, we present a canonical labeling algorithm for Latin squares of order that runs in average time bounded by a polynomial in . The algorithm can be used to solve isomorphism problems for many combinatorial objects that can be encoded using Latin squares, including quasigroups, Steiner triple systems, Mendelsohn triple systems, 1‐factorizations, nets, affine planes, and projective planes.

The flower intersection problem for Kirkman triple systems

It is known that in anyr‐coloring of the edges of a completer‐uniform hypergraph, there exists a spanning monochromatic component. Given a Steiner triple system onnvertices, what is the largest monochromatic component one can guarantee in an arbitrary 3‐coloring of the edges? Gyárfás proved that is an absolute lower bound and that this lower bound is best possible for infinitely many . On the other hand, we prove that for almost all Steiner triple systems the lower bound is actually . We obtain this result as a consequence of a more general theorem which shows that the lower bound depends on the size of a largest 3‐partite hole(ie, disjoint sets with such that no edge intersects all of ) in the Steiner triple system (Gyárfás previously observed that the upper bound depends on this parameter). Furthermore, we show that this lower bound is tight unless the structure of the Steiner triple system and the coloring of its edges are restricted in a certain way. We also suggest a variety of other Ramsey problems in the setting of Steiner triple systems.

Some results on quadrilaterals in Steiner triple systems

Population II stars are old, metal-poor, Galactic halo stars with high proper motion. We have carried out a visual binary survey of 164 halo stars in the solar neighborhood (median distance 100 pc), using infrared speckle interferometry, adaptive optics, and wide field direct imaging. The sample is based on the lists of Population II stars of Carney et al. (1994) and Norris (1986), with reliable distances from HIPPARCOS measurements.At face value, we found 33 binaries, 6 triples, and 1 quadruple system. When we limit ourselves to K-band flux ratios larger than 0.1 (to avoid background contamination), the numbers drop to 9 binaries and 1 triple, corresponding to a binary frequency of 6–7% above our angular resolution limit of about 0.1 arcsec. If we count all systems with K-band flux ratios greater than 0.01, we obtain 15 more binaries and 3 more triples, corresponding to a binary frequency for projected separations in excess of 10 AU of around 20 %. This is to be compared with the frequency of spectroscopic binaries (up to a period of 3000 days) of Population II stars of about 15 % (Latham et al. 2002). We also determined a semi-major axis distribution for our visual Population II binary and triple systems, which appears to be remarkably different from that of Population I stars. Second epoch-observations must help confirm the reality of our results.

The statistics of catalogued quadruple stars consisting of two binaries (hierarchy 2+2) is studied in comparison with triple stars, with respective sample sizes of 81 and 724. Seven representative quadruple systems are discussed in greater detail. The properties of multiple stars do not correspond to the products of dynamical decay of small clusters, hence the N-body dynamics is not the dominant process of their formation. On the other hand, rotationally-driven (cascade) fragmentation possibly followed by migration of inner and/or outer orbits to shorter periods is a promising scenario to explain the origin of triple and quadruple stars. Our main results are: (i) Quadruple systems of Epsilon Lyr type with similar masses and inner periods are common. (ii) The distributions of the inner periods in triple and quadruple stars are similar and bimodal. The inner mass ratios do not correlate with the inner periods. (iii) The statistics of outer periods and mass ratios in triples and quadruples are different. The median outer mass ratio in triples is 0.39 independently of the outer period, which has a smooth distribution. In contrast, the outer periods of 25% quadruples concentrate in the narrow range from 10yr to 100yr, the outer mass ratios of these tight quadruples are above 0.6 and their two inner periods are similar to each other. (iv) The outer and inner mass ratios in triple and quadruple stars are not mutually correlated. (v) The inner and outer orbital angular momenta and periods in triple and quadruple systems with inner periods above 30d show some correlation, the ratio of outer-to-inner periods is mostly comprised between 5 and 10^4. In the systems with small period ratios the directions of the orbital spins are correlated, while in the systems with large ratios they are not.

A minimal defining set of a Steiner triple system on a points (STS(v)) is a partial Steiner triple system contained in only this STS(v), and such that any of its proper subsets is contained in at least two distinct STS(v)s. We consider the standard doubling and tripling constructions for STS(2v + 1) and STS(3v) from STS(v) and show how minimal defining sets of an STS(v) gives rise to minimal defining sets in the larger systems. We use this to construct some new families of defining sets. For example, for Steiner triple systems on, 3 points; we construct minimal defining sets of volumes varying by as much as 7(n-/-).