The Gauss-Bonnet theorem for vector bundles

We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle E of even rank over a closed compact orientable manifold M. This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special case when M is a Riemannian manifold and E is the tangent bundle of M endowed with the Levi-Civita connection. The proof is based on an explicit geometric construction of the Thom class for 2-plane bundles. Dedicated to the memory of Philip Bell Research partially supported by NSF grant DMS-9703852.     

Solution of an extremal problem for sets using resultants of polynomials

A new, short proof is given of the following theorem of Bollobás: LetA ₁,..., A_h andB ₁,..., B_h be collections of sets with _i ¦A _i¦=r,¦B_i¦=s and ¦A _iB_j¦=Ø if and only ifi=j, thenh( _s ^r+s ). The proof immediately extends to the generalizations of this theorem obtained by Frankl, Alon and others.     

Temperely-Lieb Algebras and the Four-Color Theorem

The Temperley–Lieb algebra T_n with parameter 2 is the associative algebra over Q generated by 1,e₀,e₁, . . .,e_n, where the generators satisfy the relations if |i–j|=1 and e_ie_j=e_je_i if |i–j|2. We use the Four Color Theorem to give a necessary and sufficient condition for certain elements of T_n to be nonzero. It turns out that the characterization is, in fact, equivalent to the Four Color Theorem.* Partially supported by NSF under Grant DMS-9802859 and by NSA under grant MDA904-97-1-0015. Partially supported by NSF under Grant DMS-9623031 and by NSA under Grant MDA904-98-1-0517.     

On the average hyperoscillations of planted plane trees

Assume that the leaves of a planted plane tree are enumerated from left to right by 1, 2, .... Thej-ths-turn of the tree is defined to be the root of the (unique) subtree of minimal height with leavesj, j+1, ...,j+s−1. If all trees withn nodes are regarded equally likely, the average level number of thej-ths-turn tends to a finite limitα _s (j), which is of orderj ^1/2. Thej-th ”s-hyperoscillation”α ₁(j)−α _s+1(j) is given by 1/2α ₁(s)+O(j ^−1/2) and therefore tends (forj → ∞) to a constant behaving like √8/π·s ^1/2 fors → ∞. These results are obtained by setting up appropriate generating functions, which are expanded about their (algebraic) singularities nearest to the origin, so that the asymptotic formulas are consequences of the so-called Darboux-Pólyamethod.     

On the Crossing Number of without Computer Assistance

We present several general results about drawings of , as a beginning to trying to determine its crossing number. As application, we give a complete proof that the crossing number of K₉ is 36 and that all drawings in one large, natural class of drawings of K₁₁ have at least 100 crossings.     

On A Hypergraph Turán Problem Of Frankl

Let be the 2k-uniform hypergraph obtained by letting P₁, . . .,P_r be pairwise disjoint sets of size k and taking as edges all sets P_i∪P_j with i ≠ j. This can be thought of as the ‘k-expansion’ of the complete graph K_r: each vertex has been replaced with a set of size k. An example of a hypergraph with vertex set V that does not contain can be obtained by partitioning V = V1 ∪V₂ and taking as edges all sets of size 2k that intersect each of V₁ and V₂ in an odd number of elements. Let denote a hypergraph on n vertices obtained by this construction that has as many edges as possible. For n sufficiently large we prove a conjecture of Frankl, which states that any hypergraph on n vertices that contains no has at most as many edges as . Sidorenko has given an upper bound of for the Tur′an density of for any r, and a construction establishing a matching lower bound when r is of the form 2^p+1. In this paper we also show that when r=2^p+1, any -free hypergraph of density looks approximately like Sidorenko’s construction. On the other hand, when r is not of this form, we show that corresponding constructions do not exist and improve the upper bound on the Turán density of to , where c(r) is a constant depending only on r. The backbone of our arguments is a strategy of first proving approximate structure theorems, and then showing that any imperfections in the structure must lead to a suboptimal configuration. The tools for its realisation draw on extremal graph theory, linear algebra, the Kruskal–Katona theorem and properties of Krawtchouck polynomials. * Research supported in part by NSF grants DMS-0355497, DMS-0106589, and by an Alfred P. Sloan fellowship.     

Cutting planes,connectivity, and threshold logic

Originating from work in operations research the cutting plane refutation systemCP is an extension of resolution, where unsatisfiable propositional logic formulas in conjunctive normal form are recognized by showing the non-existence of boolean solutions to associated families of linear inequalities. Polynomial sizeCP proofs are given for the undirecteds-t connectivity principle. The subsystemsCP _q ofCP, forq2, are shown to be polynomially equivalent toCP, thus answering problem 19 from the list of open problems of [8]. We present a normal form theorem forCP ₂-proofs and thereby for arbitraryCP-proofs. As a corollary, we show that the coefficients and constant terms in arbitrary cutting plane proofs may be exponentially bounded by the number of steps in the proof, at the cost of an at most polynomial increase in the number of steps in the proof. The extensionCPLE ⁺, introduced in [9] and there shown top-simulate Frege systems, is proved to be polynomially equivalent to Frege systems. Lastly, since linear inequalities are related to threshold gates, we introduce a new threshold logic and prove a completeness theorem.Supported in part by NSF grant DMS-9205181 and by US-Czech Science and Technology Grant 93-205Partially supported by NSF grant CCR-9102896 and by US-Czech Science and Technology Grant 93-205     

Drawing 4-Pfaffian graphs on the torus

We say that a graph G is k-Pfaffian if the generating function of its perfect matchings can be expressed as a linear combination of Pfaffians of k matrices corresponding to orientations of G. We prove that 3-Pfaffian graphs are 1-Pfaffian, 5-Pfaffian graphs are 4-Pfaffian and that a graph is 4-Pfaffian if and only if it can be drawn on the torus (possibly with crossings) so that every perfect matching intersects itself an even number of times. We state conjectures and prove partial results for k>5. The author was supported in part by NSF under Grant No. DMS-0200595 and DMS-0701033.     

Complete Minors In <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">s,s</Emphasis></Subscript>-Free Graphs

We prove that for a fixed integer s2 every K _s,s -free graph of average degree at least r contains a K _p minor where . A well-known conjecture on the existence of dense K _s,s -free graphs would imply that the value of the exponent is best possible. Our result implies Hadwigers conjecture for K _s,s -free graphs whose chromatic number is sufficiently large compared with s.     

Note making a <Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>-free graph bipartite

We show that every K ₄-free graph G with n vertices can be made bipartite by deleting at most n ²/9 edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete 3-partite graph with parts of size n/3. This proves an old conjecture of P. Erdős. Research supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant, and by an Alfred P. Sloan fellowship.     

An approximate Dirac-type theorem for k-uniform hypergraphs

A k-uniform hypergraph is hamiltonian if for some cyclic ordering of its vertex set, every k consecutive vertices form an edge. In 1952 Dirac proved that if the minimum degree in an n-vertex graph is at least n/2 then the graph is hamiltonian. We prove an approximate version of an analogous result for uniform hypergraphs: For every K ≥ 3 and γ > 0, and for all n large enough, a sufficient condition for an n-vertex k-uniform hypergraph to be hamiltonian is that each (k − 1)-element set of vertices is contained in at least (1/2 + γ)n edges. Research supported by NSF grant DMS-0300529. Research supported by KBN grant 2P03A 015 23 and N201036 32/2546. Part of research performed at Emory University, Atlanta. Research supported by NSF grant DMS-0100784.     

The chromatic number of random graphs at the double-jump threshold

A crucial step in the Erdös-Rényi (1960) proof that the double-jump threshold is also the planarity threshold for random graphs is shown to be invalid. We prove that whenp=1/n, almost all graphs do not contain a cycle with a diagonal edge, contradicting Theorem 8a of Erdös and Rényi (1960). As a consequence, it is proved that the chromatic number is 3 for almost all graphs whenp=1/n.Research supported U.S. National Science Foundation Grants DMS-8303238 and DMS-8403646. The research was conducted on an exchange visit by Professor Wierman to Poland supported by the national Academy of Sciences of the USA and the Polish Academy of Sciences.     

Curvature and uniformization

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu-e ^2u=K_o(z) on general open surfaces. A few other topics are discussed, including boundary behavior of the conformal factore ^2u giving the Poincaré metric when the Riemann surface has smoothly bounded compact closure, and also a curvature equation proof of Koebe's disk theorem. Research supported in part by NSF Grant DMS-9971975 and also at MSRI by NSF grant DMS-9701755. Research supported in part by NSF Grant DMS-9877077     

On disjointly representable sets

A system of setsE ₁,E ₂, ...,E _k ⊂X is said to be disjointly representable if there existx ₁,x ₂, ...,x _k teX such thatx _i teE _j ⇔i=j. Letf(r, k) denote the maximal size of anr-uniform set-system containing nok disjointly representable members. In the first section the exact value off(r, 3) is determined and (asymptotically sharp) bounds onf(r, k),k>3 are established. The last two sections contain some generalizations, in particular we prove an analogue of Sauer’ theorem [16] for uniform set-systems. Dedicated to Paul Erdős on his seventieth birthday     

Every large set of equidistant (0, +1, −1)-vectors forms a sunflower

A theorem of Deza asserts that ifH ₁, ...,H _m ares-sets any pair of which intersects in exactlyd elements and ifm ≧s ² −s+2, then theH _i form aΔ-system, i.e. . In other words, every large equidistant (0, 1)-code of constant weight is trivial. We give a (0, +1, −1) analogue of this theorem.     

Thep-intersection number of a complete bipartite graph and orthogonal double coverings of a clique

Thep-intersection graph of a collection of finite sets {S _i } _i=1 ⁿ is the graph with vertices 1, ...,n such thati, j are adjacent if and only if |S _i S _j |p. Thep-intersection number of a graphG, herein denoted _p (G), is the minimum size of a setU such thatG is thep-intersection graph of subsets ofU. IfG is the complete bipartite graphK _n,n andp2, then _p (K _{n, n} )(n ²+(2p–1)n)/p. Whenp=2, equality holds if and only ifK _n has anorthogonal double covering, which is a collection ofn subgraphs ofK _n , each withn–1 edges and maximum degree 2, such that each pair of subgraphs shares exactly one edge. By construction,K _n has a simple explicit orthogonal double covering whenn is congruent modulo 12 to one of {1, 2, 5, 7, 10, 11}.Research supported in part by ONR Grant N00014-5K0570.     

Complete multipartite decompositions of complete graphs and complete <Emphasis Type="Italic">n</Emphasis>-partite graphs

In this paper, a new concept of an optimal complete multipartite decomposition of type 1 (type 2) of a complete n-partite graph Q _n is proposed and another new concept of a normal complete multipartite decomposition of K _n is introduced. It is showed that an optimal complete multipartite decomposition of type 1 of K _n is a normal complete multipartite decomposition. As for any complete multipartite decomposition of K _n , there is a derived complete multipartite decomposition for Q _n . It is also showed that any optimal complete multipartite decomposition of type 1 of Q _n is a derived decomposition of an optimal complete multipartite decomposition of type 1 of K _n . Besides, some structural properties of an optimal complete multipartite decomposition of type 1 of K _n are given. Supported by the National Natural Science Foundation of China (10271110).     

Sharp Bounds For Some Multicolor Ramsey Numbers

Let H ₁,H ₂, . . .,H _k+1 be a sequence of k+1 finite, undirected, simple graphs. The (multicolored) Ramsey number r(H ₁,H ₂,...,H _k+1) is the minimum integer r such that in every edge-coloring of the complete graph on r vertices by k+1 colors, there is a monochromatic copy of H _i in color i for some 1ik+1. We describe a general technique that supplies tight lower bounds for several numbers r(H ₁,H ₂,...,H _k+1) when k2, and the last graph H _k+1 is the complete graph K _m on m vertices. This technique enables us to determine the asymptotic behaviour of these numbers, up to a polylogarithmic factor, in various cases. In particular we show that r(K ₃,K ₃,K _m ) = (m ³ poly logm), thus solving (in a strong form) a conjecture of Erdos and Sós raised in 1979. Another special case of our result implies that r(C ₄,C ₄,K _m ) = (m ² poly logm) and that r(C ₄,C ₄,C ₄,K _m ) = (m ²/log² m). The proofs combine combinatorial and probabilistic arguments with spectral techniques and certain estimates of character sums.* Research supported in part by a State of New Jersey grant, by a USA Israeli BSF grant and by a grant from the Israel Science Foundation. Research supported by NSF grant DMS 9704114.     

Note on a conjecture of toft

A conjecture of Toft [17] asserts that any 4-critical graph (or equivalently, every 4-chromatic graph) contains a fully odd subdivision ofK ₄. We show that if a graphG has a degree three nodev such thatG-v is 3-colourable, then eitherG is 3-colourable or it contains a fully oddK ₄. This resolves Toft's conjecture in the special case where a 4-critical graph has a degree three node, which is in turn used to prove the conjecture for line-graphs. The proof is constructive and yields a polynomial algorithm which given a 3-degenerate graph either finds a 3-colouring or exhibits a subgraph that is a fully odd subdivision ofK ₄. (A graph is 3-degenerate if every subgraph has some node of degree at most three.)