报告题目1：Some new results on the linear Turán problems of hypergraphs
报告摘要：An r-uniform hypergraph is linear if every two edges intersect in at most one vertex. Given a family of r-uniform hypergraph F, the linear Turán number exrlin(n,F) is the maximum number of edges of a linear r-uniform hypergraph on n vertices that does not contain any member of F as a subhypergraph. Let F be a graph and r ≥ 3 a positive integer. The r-expansion of F is the r-graph F+ obtained from F by enlarging each edge of F with r − 2 new vertices disjoint from V(F) such that distinct edges of F are enlarged by distinct vertices. In this talk, we first focus on the linear Turán problem of the bipartite hypergraph K+s,t, and present some new bounds for the linear Turán number of K+s,t for t ≥ s ≥ 2. Then we develop a new tool to establish the connection between spectral radius of the adjacency tensor and structural properties of a hypergraph, and prove that when n is sufficiently large, the spectral radius ρ(K+4) of the adjacency tensor of K+4 is no more than n/3 , i.e., ρ(K+4)≤ n/3 , with equality if and only if 3|n and H is a transversal design, where K+4 is the 3-uniform hypergraph obtained from K4 by enlarging each edge of K4 with a new vertex. An immediate corollary of this result is that ex3lin(n, K+4 ) = n2/9 for sufficiently large n and 3|n. This is the joint work with Guorong Gao, and Yuan Hou.
报告题目2：JUMPING DENSITIES OF HYPERGRAPHS
摘要: A real number α∈[0,1) is a jump for an integer r ≥ 2 if there exists c > 0 such that no number in (α,α+c) can be the Turán density of a family of r-uniform graphs. A classical result of Erdős and Stone implies that that every number in [0,1) is a jump for r = 2. Erdős also showed that every number in [0, r!/rr ) is a jump for r ≥ 3 and asked whether every number in [0,1) is a jump for r ≥ 3. Frankl and Rödl gave a negative answer by showing a sequence of non-jumps for every r ≥ 3. After this, Erdős modified the question to be whether r!/rr is a jump for r ≥ 3? What’s the smallest non-jump? Frankl, Peng, Rödl and Talbot showed that 5r!/2rr is a non-jump for r ≥ 3. However, Baber and Talbot showed that there are more jumps by proving that every α∈[0.2299,0.2316)∪[0.2871,8/27) is a jump for r =3. But Pikhurko showed that the set of non-jumps has the cardinality of the continuum by proving that the set of all possible Turán densities of r-uniform graphs has the cardinality of the continuum for r ≥ 3. However, whether r!/rr is a jump for r ≥ 3 remains open, and 5r!/2rr has remained the known smallest non-jump for r ≥ 3. We give a smaller non-jump by showing that 54r!/25rr is a non-jump for r ≥ 3. Furthermore, we give infinitely many irrational non-jumps for every r ≥ 3. This is a joint work with Zilong Yan.
专家简介：彭岳建，现为湖南大学数学学院教授，博士生导师，1989年获湘潭大学数学学士学位，1992年硕士毕业于复旦大学数学系，2001年获Emory大学（美国）数学博士。主要研究方向为极值图论，在JCTB, JCTA, CPC，SIDA等知名期刊发表论文50余篇， 主持国家自然科学基金面上项目和重点项目。