数学研究所系列学术报告（常安 福州大学；彭岳建 湖南大学）

报告题目1：Some new results on the linear Turán problems of hypergraphs

报告专家：常安教授（福州大学）

报告时间：12月14日周二14：30-15：30

腾讯会议号： 483-632-013

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 ex_r^lin(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 ex₃^lin(n, K⁺₄ ) = n²/9 for sufficiently large n and 3|n. This is the joint work with Guorong Gao, and Yuan Hou.

专家简介： 常安，福州大学教授，博士生导师。1998年6月毕业于四川大学应用数学专业，获博士学位。目前兼任中国运筹会理事、中国数学会组合数学与图论专业委员会副主任（常务理事）。主要从事图论领域中图与超图的谱理论及应用研究。已在国内外专业期刊发表研究论文60多篇，参加了包括国家重点研究计划（973计划）项目课题、国家自然科学基金重点项目在内的十余项国家级科研项目的研究工作，并承担了多项包括国家自然科学基金面上项目和其他省级科研项目的研究工作。2004年获福建省科学技术二等奖。

报告题目2：JUMPING DENSITIES OF HYPERGRAPHS

报告专家：彭岳建教授（湖南大学）

报告时间：12月14日周二15:30 -16:30

腾讯会议号：483-632-013

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!/r^r ) 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!/r^r is a jump for r ≥ 3? What’s the smallest non-jump? Frankl, Peng, Rödl and Talbot showed that 5r!/2r^r 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!/r^r is a jump for r ≥ 3 remains open, and 5r!/2r^r has remained the known smallest non-jump for r ≥ 3. We give a smaller non-jump by showing that 54r!/25r^r 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。

邀请人：数学研究所