Effective magnetic moments of a Heisenberg linear chaing

The effective magnetic moments of a linear chain of up to 12 spin-1/2 particles interacting through the isotropic Heisenberg coupling are computed exactly over the entire temperature range of interest.     

Counting substructures I: Color critical graphs

Let F be a graph which contains an edge whose deletion reduces its chromatic number. We prove tight bounds on the number of copies of F in a graph with a prescribed number of vertices and edges. Our results extend those of Simonovits (1968) [8], who proved that there is one copy of F, and of Rademacher, Erd?s (1962) [1] and [2] and Lovász and Simonovits (1983) [4], who proved similar counting results when F is a complete graph.One of the simplest cases of our theorem is the following new result. There is an absolute positive constant c such that if n is sufficiently large and 1?q<cn, then every n vertex graph with ⌊n²/4⌋+q edges contains at least

     

A new short proof of a theorem of Ahlswede and Khachatrian

Ahlswede and Khachatrian [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121-138] proved the following theorem, which answered a question of Frankl and Füredi [P. Frankl, Z. Füredi, Nontrivial intersecting families, J. Combin. Theory Ser. A 41 (1986) 150-153]. Let 2?t+1?k?2t+1 and n?(t+1)(k−t+1). Suppose that F is a family of k-subsets of an n-set, every two of which have at least t common elements. If |_?F∈FF|<t, then , and this is best possible. We give a new, short proof of this result. The proof in [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121-138] requires the entire machinery of the proof of the complete intersection theorem, while our proof uses only ordinary compression and an earlier result of Wilson [R.M. Wilson, The exact bound in the Erd?s-Ko-Rado theorem, Combinatorica 4 (1984) 247-257].     

An intersection theorem for four sets

Fix integers n,r?4 and let F denote a family of r-sets of an n-element set. Suppose that for every four distinct A,B,C,D∈F with |A∪B∪C∪D|?2r, we have A∩B∩C∩D≠∅. We prove that for n sufficiently large, , with equality only if _?F∈FF≠∅. This is closely related to a problem of Katona and a result of Frankl and Füredi [P. Frankl, Z. Füredi, A new generalization of the Erd?s-Ko-Rado theorem, Combinatorica 3 (3-4) (1983) 341-349], who proved a similar statement for three sets. It has been conjectured by the author [D. Mubayi, Erd?s-Ko-Rado for three sets, J. Combin. Theory Ser. A, 113 (3) (2006) 547-550] that the same result holds for d sets (instead of just four), where d?r, and for all n?dr/(d−1). This exact result is obtained by first proving a stability result, namely that if |F| is close to then F is close to satisfying _?F∈FF≠∅. The stability theorem is analogous to, and motivated by the fundamental result of Erd?s and Simonovits for graphs.     

An Explicit Construction for a Ramsey Problem

An explicit coloring of the edges of K_n is constructed such that every copy of K₄ has at least four colors on its edges. As n , the number of colors used is n^1/2+o(1). This improves upon the previous bound of O(n^2/3) due to Erds and Gyárfás obtained by probabilistic methods. The exponent 1/2 is optimal, since it is known that at least (n^1/2) colors are required in such a coloring.The coloring is related to constructions giving lower bounds for the multicolor Ramsey number r_k(C₄). It is more complicated however, because of restrictions imposed on interactions between color classes.* Research supported in part by NSF Grant No. DMS–9970325.     

Almost all triangle-free triple systems are tripartite

A triangle in a triple system is a collection of three edges isomorphic to {123,124,345}. A triple system is triangle-free if it contains no three edges forming a triangle. It is tripartite if it has a vertex partition into three parts such that every edge has exactly one point in each part. It is easy to see that every tripartite triple system is triangle-free. We prove that almost all triangle-free triple systems with vertex set [n] are tripartite. Our proof uses the hypergraph regularity lemma of Frankl and R?dl [13], and a stability theorem for triangle-free triple systems due to Keevash and the second author [15].     

Multicolor Ramsey Numbers For Complete Bipartite Versus Complete Graphs

Let be graphs. The multicolor Ramsey number is the minimum integer r such that in every edge‐coloring of by k colors, there is a monochromatic copy of in color i for some . In this paper, we investigate the multicolor Ramsey number , determining the asymptotic behavior up to a polylogarithmic factor for almost all ranges of t and m. Several different constructions are used for the lower bounds, including the random graph and explicit graphs built from finite fields. A technique of Alon and Rödl using the probabilistic method and spectral arguments is employed to supply tight lower bounds. A sample result is for any t and m, where c₁ and c₂ are absolute constants.     

Kinetic study of copolymerization of linalool and methyl methacrylate initiated by selenonium ylide

Free radical copolymerization of an acyclic monoterpenoid linalool (LIN) and methyl methacrylate (MMA) in dioxan was carried out in dilatometer under an inert atmosphere of nitrogen for 90 min at 60 ± 1°C by using diphenyl selenonium 2,3,4,5‐tetraphenylcyclopentadienylide (selenonium ylide) as an initiator. The kinetic expression of the reaction is R_p ∝ [ylide]^0.5[MMA]^1.0[LIN]^1.0. The activation energy of copolymerization was estimated to be 43.7 kJ mol^?1. The formation of a functional copolymer is evidenced by spectral analysis. The copolymer was characterized by FTIR, ¹H NMR, ¹³C NMR, DSC, and TGA analysis. © 2010 Wiley Periodicals, Inc. Int J Chem Kinet 43: 43–52, 2011