On Frankl and Füredi’s conjecture for 3-uniform hypergraphs

Frankl and Füredi in [1] conjectured that the r-graph with m edges formed by taking the first m sets in the colex ordering of N^(r) has the largest Lagrangian of all r-graphs with m edges. Denote this r-graph by C _r,m and the Lagrangian of a hypergraph by λ(G). In this paper, we first show that if \(\leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3 \end{array}} \right)\), G is a left-compressed 3-graph with m edges and on vertex set [t], the triple with minimum colex ordering in G ^c is (t ? 2 ? i)(t ? 2)t, then λ(G) ≤ λ(C _3,m ). As an implication, the conjecture of Frankl and Füredi is true for \(\left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right) - 6 \leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right)\).     

On Hua-Tuan’s conjecture

Let G be a finite group and |G| = pn, p be a prime. For 0 m n, sm(G) denotes the number of subgroups of of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan have ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1 + p, 1 + p + p2 or 1 + p + 2p2 (mod p3). In this paper, we investigate the conjecture, and give some p-groups in which the conjecture holds and some examples in which the conjecture does not hold.     

Taylor’s modularity conjecture holds for linear idempotent varieties

The “Modularity Conjecture” is the assertion that the join of two nonmodular varieties in the lattice of interpretability types is nonmodular. We establish the veracity of this conjecture for the case of linear idempotent varieties. We also establish analogous results concerning n-permutability for some n, and the satisfaction of nontrivial congruence identities.     

On De Jong’s conjecture

Let X be a smooth projective curve over a finite field F _q . Let ρ be a continuous representation π(X) → GL _n (F), where F = F _l ((t)) with F _l being another finite field of order prime to q. Assume that is irreducible. De Jong’s conjecture says that in this case is finite. As was shown in the original paper of de Jong, this conjecture follows from an existence of an F-valued automorphic form corresponding to ρ is the sense of Langlands. The latter follows, in turn, from a version of the Geometric Langlands conjecture. In this paper we sketch a proof of the required version of the geometric conjecture, assuming that char(F) ≠ 2, thereby proving de Jong’s conjecture in this case.     

On Hua-Tuan’s conjecture Ⅱ

Let G be a group of order pn, p a prime. For 0 m n, sm(G) denotes the number of subgroups of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan had ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1+p, 1+p+p2 or 1+p+2p2(mod p3). The conjecture has a negative answer. In this paper, we further investigate the conjecture and propose two new conjectures.     

On Ueno’s conjecture K

We show that if X is a smooth complex projective variety with Kodaira dimension 0 then the Kodaira dimension of a general fiber of its Albanese map is at most . J. A. Chen was partially supported by NCTS, TIMS, and NSC of Taiwan. C. D. Hacon was partially supported by NSF research grant no: 0456363 and an AMS Centennial Scholarship. We would like to thank J. Kollár, R. Lazarsfeld, C.-H. Liu, M. Popa, P. Roberts, and A. Singh for many useful comments on the contents of this paper.     

On Sarnak’s conjecture and Veech’s question for interval exchanges

Using a criterion due to Bourgain [10] and the generalization of the self-dual induction defined in [19], for each primitive permutation we build a large family of k-interval exchanges satisfying Sarnak’s conjecture, and, for at least one permutation in each Rauzy class, smaller families for which we have weak mixing, which implies a prime number theorem, and simplicity in the sense of Veech.     

On Oliver’s p-group conjecture: II

Let p be an odd prime and S a finite p-group. B. Oliver’s conjecture arises from an open problem in the theory of p-local finite groups. It is the claim that a certain characteristic subgroup \mathfrakX(S){\mathfrak{X}(S)} of S always contains the Thompson subgroup. In previous work the first two authors and M. Lilienthal recast Oliver’s conjecture as a statement about the representation theory of the factor group S/\mathfrakX(S){S/\mathfrak{X}(S)}. We now verify the conjecture for a wide variety of groups S/\mathfrakX(S){S/\mathfrak{X}(S)}.     

Baker’s conjecture and Eremenko’s conjecture for functions with negative zeros

We introduce a new technique that allows us to make progress on two long standing conjectures in transcendental dynamics: Baker's conjecture that a transcendental entire function of order less than 1/2 has no unbounded Fatou components, and Eremenko's conjecture that all the components of the escaping set of an entire function are unbounded. We show that both conjectures hold for many transcendental entire functions whose zeros all lie on the negative real axis, in particular those of order less than 1/2. Our proofs use a classical distortion theorem based on contraction of the hyperbolic metric, together with new results which show that the images of certain curves must wind many times round the origin.     

Hadwiger’s conjecture for uncountable graphs

We consider the infinite form of Hadwiger’s conjecture. We give a(n apparently novel) proof of Halin’s 1967 theorem stating that every graph X with coloring number \(>\kappa \) (specifically with chromatic number \(>\kappa \)) contains a subdivision of \(K_\kappa \). We also prove that there is a graph of cardinality \(2^\kappa \) and chromatic number \(\kappa ^+\) which does not contain \(K_{\kappa ^+}\) as a minor. Further, it is consistent that every graph of size and chromatic number \(\aleph _1\) contains a subdivision of \(K_{\aleph _1}\).     

A Comment on Ryser’s Conjecture for Intersecting Hypergraphs

Let \(\tau({\mathcal{H}})\) be the cover number and \(\nu({\mathcal{H}})\) be the matching number of a hypergraph \({\mathcal{H}}\). Ryser conjectured that every r-partite hypergraph \({\mathcal{H}}\) satisfies the inequality \(\tau({\mathcal{H}}) \leq (r-1) \nu ({\mathcal{H}})\). This conjecture is open for all r ≥ 4. For intersecting hypergraphs, namely those with \(\nu({\mathcal{H}}) = 1\), Ryser’s conjecture reduces to \(\tau({\mathcal{H}}) \leq r-1\). Even this conjecture is extremely difficult and is open for all r ≥ 6. For infinitely many r there are examples of intersecting r-partite hypergraphs with \(\tau({\mathcal{H}}) = r-1\), demonstrating the tightness of the conjecture for such r. However, all previously known constructions are not optimal as they use far too many edges. How sparse can an intersecting r-partite hypergraph be, given that its cover number is as large as possible, namely \(\tau({\mathcal{H}}) \ge r-1\)? In this paper we solve this question for r ≤ 5, give an almost optimal construction for r = 6, prove that any r-partite intersecting hypergraph with τ(H) ≥ r ? 1 must have at least \((3-\frac{1}{\sqrt{18}})r(1-o(1)) \approx 2.764r(1-o(1))\) edges, and conjecture that there exist constructions with Θ(r) edges.     

Vaught’s conjecture for some meager groups

AssumeG is a superstable group ofM-rank 1 and the division ring of pseudo-endomorphisms ofG is a prime field. We prove a relative Vaught’s conjecture for Th(G). When additionallyU(G) =ω, this yields Vaught’s conjecture for Th(G). Research supported by KBN grant 2 P03A 006 09.     

Coleman-Oort’s conjecture for degenerate irreducible curves

We state and prove the analogue of a conjecture of Coleman and Oort for the locus of degenerate irreducible curves.