A polynomial resultant approach to algebraic construction of extremal graphs

The Turán problem asks for the largest number of edges ex(n,H) in an n-vertex graph not containing a fixed forbidden subgraphH, which is one of the most important problems in extremal graph theory. However the order of magnitude of ex(n,H) for bipartite graphs is known only in a handful of cases. In particular, giving explicit constructions of extremal graphs is very challenging in this field. In this paper, we develop a polynomail resultant approach to algebraic construction of explicit extremal graphs, which can efficiently decide whether a specified structure exists. A key insight in our approach is the multipolynomial resultant, which is a fundamental tool of computational algebraic geometry. Our main results include the matched lowers bounds for Turán number of 1-subdivision of K3,t1 and linear Turán number of Berge theta hyerpgraph Θ3,t2 with t1 = 25 and t2 = 217. Moreover, the constant t1 improves the random algebraic construction of Bukh and Conlon [Rational exponents in extremal graph theory, J. Eur. Math. Soc. 20 (2018), 1747-1757] and makes progress on the known estimation for the smallest value of t1 concerning a problem posed by Conlon, Janzer and Lee [More on the extremal number of subdivisions, Combinatorica, to appear], while the constant t2 improves a result of He and Tait [Hypergraphs with few berge paths of fixed length between vertices, SIAM J. Discrete Math., 33(3), 1472-1481].

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