arXiv subjects

Oliver Janzer

Publications and source records attributed to Oliver Janzer.

46 records · Page 3Linked to original sources

arXiv2021

On the Zarankiewicz problem for graphs with bounded VC-dimension

The problem of Zarankiewicz asks for the maximum number of edges in a bipartite graph on $n$ vertices which does not contain the complete bipartite graph $K_{k,k}$ as a subgraph. A classical theorem due to Kővári, Sós, and Turán says that this number of edges is $O\left(n^{2 - 1/k}\right)$. An important variant of this problem is the analogous question in bipartite graphs with VC-dimension at most $d$, where $d$ is a fixed integer such that $k \geq d \geq 2$. A remarkable result of Fox, Pach, Sheffer, Suk, and Zahl [J. Eur. Math. Soc. (JEMS), no. 19, 1785-1810] with multiple applications in incidence geometry shows that, under this additional hypothesis, the number of edges in a bipartite graph on $n$ vertices and with no copy of $K_{k,k}$ as a subgraph must be $O\left(n^{2 - 1/d}\right)$. This theorem is sharp when $k=d=2$, because by design any $K_{2,2}$-free graph automatically has VC-dimension at most $2$, and there are well-known examples of such graphs with $Ω\left(n^{3/2}\right)$ edges. However, it turns out this phenomenon no longer carries through for any larger $d$. We show the following improved result: the maximum number of edges in bipartite graphs with no copies of $K_{k,k}$ and VC-dimension at most $d$ is $o(n^{2-1/d})$, for every $k \geq d \geq 3$.

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arXiv2021

Long directed paths in Eulerian digraphs

An old conjecture of Bollobás and Scott asserts that every Eulerian directed graph with average degree $d$ contains a directed cycle of length at least $Ω(d)$. The best known lower bound for this problem is $Ω(d^{1/2})$ by Huang, Ma, Shapira, Sudakov and Yuster. They asked whether this estimate can be improved at least for directed paths instead of cycles and whether one can find a long path starting from any vertex if the host digraph is connected. In this paper we break the $\sqrt{d}$ barrier, showing how to find a path of length $Ω(d^{1/2+1/40})$ from any vertex of a connected Eulerian digraph.

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arXiv2020

Coloring linear hypergraphs: the Erdős-Faber-Lovász conjecture and the Combinatorial Nullstellensatz

The long-standing Erdős-Faber-Lovász conjecture states that every $n$-uniform linear hypergaph with $n$ edges has a proper vertex-coloring using $n$ colors. In this paper we propose an algebraic framework to the problem and formulate a corresponding stronger conjecture. Using the Combinatorial Nullstellensatz, we reduce the Erdős-Faber-Lovász conjecture to the existence of non-zero coefficients in certain polynomials. These coefficients are in turn related to the number of orientations with prescribed in-degree sequences of some auxiliary graphs. We prove the existence of certain orientations, which verifies a necessary condition for our algebraic approach to work.

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arXiv2020

Polynomial bound for the partition rank vs the analytic rank of tensors

A tensor defined over a finite field $\mathbb{F}$ has low analytic rank if the distribution of its values differs significantly from the uniform distribution. An order $d$ tensor has partition rank 1 if it can be written as a product of two tensors of order less than $d$, and it has partition rank at most $k$ if it can be written as a sum of $k$ tensors of partition rank 1. In this paper, we prove that if the analytic rank of an order $d$ tensor is at most $r$, then its partition rank is at most $f(r,d,|\mathbb{F}|)$, where, for fixed $d$ and $\mathbb{F}$, $f$ is a polynomial in $r$. This is an improvement of a recent result of the author, where he obtained a tower-type bound. Prior to our work, the best known bound was an Ackermann-type function in $r$ and $d$, though it did not depend on $\mathbb{F}$. It follows from our results that a biased polynomial has low rank; there too we obtain a polynomial dependence improving the previously known Ackermann-type bound. A similar polynomial bound for the partition rank was obtained independently and simultaneously by Milićević.

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arXiv2020

More on the extremal number of subdivisions

Given a graph $H$, the extremal number $\mathrm{ex}(n,H)$ 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 $\mathrm{ex}(n, K'_{s,t}) = O(n^{3/2 - \frac{1}{2s}})$. 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 $s, k \geq 1$, we show that $\mathrm{ex}(n, L) = Θ(n^{1 + \frac{s}{sk+1}})$ 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 $r \in (1,2)$ is realisable in the sense that $\mathrm{ex}(n,H) = Θ(n^r)$ 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 $k \geq 1$. 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 $\mathrm{ex}(n,H^{k-1}) = O(n^{1 + 1/k - δ})$, 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 $\mathrm{ex}(n, H) = o(n^{2 - 1/r})$.

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arXiv2019

The extremal number of the subdivisions of the complete bipartite graph

For a graph $F$, the $k$-subdivision of $F$, denoted $F^k$, is the graph obtained by replacing the edges of $F$ with internally vertex-disjoint paths of length $k$. In this paper, we prove that $\mathrm{ex}(n,K_{s,t}^k)=O(n^{1+\frac{s-1}{sk}})$, which is tight for $t$ sufficiently large. This settles a conjecture of Conlon--Janzer--Lee, and improves on a substantial body of work by Conlon--Janzer--Lee and Jiang--Qiu.

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arXiv2019

The extremal number of longer subdivisions

For a multigraph $F$, the $k$-subdivision of $F$ is the graph obtained by replacing the edges of $F$ with pairwise internally vertex-disjoint paths of length $k+1$. Conlon and Lee conjectured that if $k$ is even, then the $(k-1)$-subdivision of any multigraph has extremal number $O(n^{1+\frac{1}{k}})$, and moreover, that for any simple graph $F$ there exists $\varepsilon>0$ such that the $(k-1)$-subdivision of $F$ has extremal number $O(n^{1+\frac{1}{k}-\varepsilon})$. In this paper, we prove both conjectures.

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arXiv2019

The Turán number of blow-ups of trees

A conjecture of Erdős from 1967 asserts that any graph on $n$ vertices which does not contain a fixed $r$-degenerate bipartite graph $F$ has at most $Cn^{2-1/r}$ edges, where $C$ is a constant depending only on $F$. We show that this bound holds for a large family of $r$-degenerate bipartite graphs, including all $r$-degenerate blow-ups of trees. Our results generalise many previously proven cases of the Erdős conjecture, including the related results of Füredi and Alon, Krivelevich and Sudakov. Our proof uses supersaturation and a random walk on an auxiliary graph.

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arXiv2018

Low analytic rank implies low partition rank for tensors

A tensor defined over a finite field $\mathbb{F}$ has low analytic rank if the distribution of its values differs significantly from the uniform distribution. An order $d$ tensor has partition rank 1 if it can be written as a product of two tensors of order less than $d$, and it has partition rank at most $k$ if it can be written as a sum of $k$ tensors of partition rank 1. In this paper, we prove that if the analytic rank of an order $d$ tensor is at most $r$, then its partition rank is at most $f(r,d,|\mathbb{F}|)$. Previously, this was known with $f$ being an Ackermann-type function in $r$ and $d$ but not depending on $\mathbb{F}$. The novelty of our result is that $f$ has only tower-type dependence on its parameters. It follows from our results that a biased polynomial has low rank; there too we obtain a tower-type dependence improving the previously known Ackermann-type bound.

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arXiv2018

Improved bounds for the extremal number of subdivisions

Let $H_t$ be the subdivision of $K_t$. Very recently, Conlon and Lee have proved that for any integer $t\geq 3$, there exists a constant $C$ such that $\text{ex}(n,H_t)\leq Cn^{3/2-1/6^t}$. In this paper, we prove that there exists a constant $C'$ such that $\text{ex}(n,H_t)\leq C'n^{3/2-\frac{1}{4t-6}}$.

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