More on the Extremal Number of Subdivisions

Abstract

Given a graph H, the extremal number ex(n, H) is the largest number of edges in an H-free graph on n vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing K ′ s,t for the subdivision of the bipartite graph K s,t , we show that ex(n, K ′ s,t ) = O(n 3/2− 1 2s ). This proves a conjecture of Kang, Kim and Liu and is tight up to the implied constant for t sufficiently large in terms of s. Second, for any integers s, k ≥ 1, we show that ex(n, L) = Θ(n 1+ s sk+1 ) for a particular graph L depending on s and k, answering another question of Kang, Kim and Liu. This result touches upon an old conjecture of Erdős and Simonovits, which asserts that every rational number r ∈ (1, 2) is realisable in the sense that ex(n, H) = Θ(n r ) for some appropriate graph H, giving infinitely many new realisable exponents and implying that 1 + 1/k is a limit point of realisable exponents for all k ≥ 1. Writing H k for the k-subdivision of a graph H, this result also implies that for any bipartite graph H and any k, there exists δ > 0 such that ex(n, H k−1 ) = O(n 1+1/k−δ ), partially resolving a question of Conlon and Lee. Third, extending a recent result of Conlon and Lee, we show that any bipartite graph H with maximum degree r on one side which does not contain C 4 as a subgraph satisfies ex(n, H) = o(n 2−1/r ).

Some remarks on the Zarankiewicz problem

“…The first traces of this random algebraic method go back some way, to work of Matoušek [20] in discrepancy theory, but it is the variant originating with Bukh [5], and developed further by the author [8], that has proved most useful. For instance, it has led to considerable progress [6,9,13,14,15,16,17] on the celebrated rational exponents conjecture of Erdős and Simonovits [12], amongst other applications [7,10]. Our main result, a general lower bound for z(m, n; s, t) valid over a broad range of m, is another application of the random algebraic method, though in a new, arguably simpler, form that returns quantitative estimates not at present available through the application of Bukh's method.…”

Turán number of bipartite graphs with no 𝐾_{𝑡,𝑡}

“…In the proof of our main theorem, we use the following technical lemma, which can be found as Lemma 2.2 in [4]. Here, a graph G is K-almost regular if the maximum degree of G is at most K-times the minimum degree.…”

“…Despite recent progress on this topic (see, e.g., [5,13,4,11,14,15,16]), this problem remains open for t ≥ 3. Moreover the following weaker form of the above conjecture, proposed by Conlon, Janzer and Lee [4] was open as well.…”

“…For a hypergraph H, the subdivision of H is the bipartite graph H ′ whose two vertex classes are V (H) and E(H), and v ∈ V (H) and e ∈ E(H) are joined by an edge if v ∈ e. Then Conjecture 3 is equivalent to asking whether ex(n, H ′ ) = o(n 2−1/t ) for a subdivision H ′ of a t-uniform hypergraph. In [4], this conjecture is proved in the special case H is a linear hypergraph (that is, any two edges of H intersect in at most one vertex), which corresponds to the case in which the bipartite graph H is K 2,2 -free. Also, it is mentioned in [5] and [4] that Conjecture 3 holds in case H is the subdivision of the complete t-uniform hypergraph with t + 1 vertices, or the subdivision of a t-partite t-uniform hypergraph.…”

“…In [4], this conjecture is proved in the special case H is a linear hypergraph (that is, any two edges of H intersect in at most one vertex), which corresponds to the case in which the bipartite graph H is K 2,2 -free. Also, it is mentioned in [5] and [4] that Conjecture 3 holds in case H is the subdivision of the complete t-uniform hypergraph with t + 1 vertices, or the subdivision of a t-partite t-uniform hypergraph.…”