# 2018-19

Organisers: ## Term 2: Hong Liu## Term 1: Jaehoon Kim and Maryam Sharifzadeh |

#### Term 2 2018/19

Date |
Name |
Title |

Jan 11 | Peter Allen (LSE) | Optimal packings of sparse graphs |

Jan 18 | Oliver Janzer (Cambridge) | The largest $K_s$-free induced subgraph in a $K_t$-free graph |

Jan 25 | Jason Long (Cambridge) | Partial Associativity in Latin Squares |

Feb 01 | David Conlon (Oxford) | Online Ramsey numbers |

Feb 08 | Sean Prendiville (Manchester) | Nonlinear problems in arithmetic Ramsey theory |

Feb 15 | Mykhaylo Tyomkyn (Oxford) | The Brown-Erdos-Sos conjecture in groups |

Feb 22 | Felix Joos (Birmingham) | Restricted subgraph embeddings |

Mar 01 | Marius Tiba (Cambridge) | Judiciously $3$-partitioning $3$-uniform hypergraphs |

Mar 07 |
Katherine Staden (Oxford) | (Please note the unusual time and location) |

Mar 08 |
Max Pitz (Hamburg) | Eulerian walks in infinite graphs |

Mar 15 | Adam Wagner (ETH) | An extremal problem concerning projective cubes |

#### Term 1 2018/19

Date |
Name |
Title |

Oct 05 | Natalie Behague (QMUL) | Hypergraph saturation irregularities |

Oct 12 | John Sylvester (Cambridge) | The dispersion time of random walks on finite graphs |

Oct 19 | António Girão (Birmingham) | A large number of m-coloured complete infinite subgraphs |

Oct 26 | Andreas Galanis (Oxford) | Phase transitions of the independence polynomial |

Nov 02 | Joel Larsson (Warwick) | The Minimum Matching problem and a local exploration game |

Nov 09 | Lorenzo Federico (Warwick) | Almost connected configuration model |

Nov 13 |
Jan Grebik (Czech academy of sciences) |
Weak* topology and graphons (Please note the unusual time and location) |

Nov 16 | Hong Liu (Warwick) | Polynomial Schur's theorem |

Nov 23 | Dong Yeap Kang (KAIST) | On the rational Turán exponents conjecture |

Nov 27 |
Robert Hancock (Czech academy of sciences) | Solution-free sets of integers (Please note the unusual time and location) |

Nov 30 | Andrey Kupavskii (Birmingham) | Structure of intersecting families and juntas |

##### Hypergraph saturation irregularities (Natalie Behague), 14:00, MS.04

We say that a graph $G$ is saturated with respect to some graph $F$ if $G$ doesn't contain any copies of $F$ but adding any new edge to $G$ creates some copy of $F$. The saturation number $\text{sat}(F,n)$ is the minimum number of edges an $F$-saturated graph on $n$ vertices can have. This forms an interesting counterpoint to the Turan number; the saturation number is in many ways less well-behaved. For example, Tuza conjectured that $\text{sat}(F,n)/n$ must tend to a limit as $n$ tends to infinity and this is still open. However, Pikhurko disproved a strengthening of Tuza's conjecture by finding a finite family of graphs, whose saturation number divided by $n$ does not tend to a limit. We will prove a similar result for hypergraphs and discuss some variants.

##### The dispersion time of random walks on finite graphs (John Sylvester), 14:00, MS.04

Consider two random processes on an $n$ vertex graph related to Internal diffusion-limited aggregation (IDLA). In each process $n$ particles perform independent random walks from a fixed origin until they reach an unvisited vertex, at which point they settle. In the first process only one particle moves until settling and then the next starts, in the second process all particles are released together. We study the *dispersion time* which is the time taken for the longest walk to settle.

We present a coupling which allows us to compare dispersion time across the two processes and show which is ''faster''. We show bounds on the dispersion time(s) in terms of more well studied parameters of random walks such as the cover time and mixing time. In addition we discuss how to determine the order of the dispersion time for many well known graphs such as expanders, complete binary trees, tori etc. This is joint work with Nicol\'{a}s Rivera, Alexandre Stauffer and Thomas Sauerwald.

##### A large number of m-coloured complete infinite subgraphs (António Girão), 14:00, MS.04

Ramsey Theorem states that for any positive integers $k$ and $m$, whenever the $k$-subsets of the natural numbers are coloured with $m$ colours there always exists an infinite subset $A$ of $\mathbb{N}$ such that $A^{(k)}$, the set of the $k$-subsets of $A$, is monochromatic. In 1975, Erd\H{o}s, Simonovits and S\H{o}s, started a new line of research, commonly known as Anti-Ramsey theory. The problems in this area lie at the opposite end of Ramsey theory, in here one is interested in finding large totally multicoloured (rainbow) substructures.

In this talk we are interested in understanding what happens in between these extremes. Given an edge colouring of a graph with a set of $m$ colours, we say

that the graph is $m$-coloured if all $m$ colours are used. For an $m$-colouring $\Delta$ of $\mathbb{N}^{(2)}$, the complete graph on $\mathbb{N}$, we denote by $\mathcal{F}_{\Delta}$ the set all values $\gamma$ for which there exists an infinite subset $X \subset\mathbb{N}$ such that $X^{(2)}$ is $\gamma$-coloured. Properties of this set were first studied by Erickson in 1994. Here, we are interested in estimating the minimum size of $\mathcal{F}_{\Delta}$ over all $m$-colourings $\Delta$ of $\mathbb{N}^{(2)}$. Indeed, we shall prove the following result. There exists an absolute constant $\alpha > 0$ such that for any positive integer $m \neq \left\{ {n \choose 2}+1, {n \choose 2}+2: n\geq 2\right\}$, $|\mathcal{F}_{\Delta}| \geq (1+\alpha)\sqrt{2m}$, for any $m$-colouring $\Delta$ of $\mathbb{N}^{(2)}$, thus proving a conjecture of Narayanan. This result is tight up to the order of the constant $\alpha$.

##### Phase transitions of the independence polynomial (Andreas Galanis), 14:00, MS.04

For a graph $G$, let $p_G(\lambda)$ denote the independence polynomial of $G$ with parameter $\lambda$. For which $\lambda$ can we approximate the value $p_G(\lambda)$ on graphs $G$ of maximum degree $\Delta$?

Recently, there have been significant developments in designing approximation algorithms for this problem by examining the location of zeros of the polynomial in the complex plane. In this talk, we will review these developments and establish barriers to the new methods coming from phase transitions in statistical physics and complex dynamics. Based on joint work with Ivona Bezakova, Leslie Ann Goldberg, and Daniel Stefankovic.

##### The Minimum Matching problem and a local exploration game (Joel Larsson), 14:00, MS.04

Consider the complete bipartite graph on $n+n$ vertices equipped with i.i.d. edge weights, and assume that their common cdf $F$ scales like $F(t)\sim t^q$ near $t=0$ for some $q>0$. The random assignment or minimum matching problem is to find a perfect matching that minimizes the total weight of all included edges. We let $M_n$ be this minimum.

The Mézard-Parisi conjecture states that the limit (in probability) of $M_n$ is $\pi^2/6$ for $q=1$, and more generally that the limit of $n^{-1+1/q} M_n$ exists. The conjecture was confirmed for $q=1$ by Aldous in ’01 and for $q>1$ by Wästlund in ’12. We extend those results to all $q>0$.

##### Almost connected configuration model (Lorenzo Federico), 14:00, MS.04

The configuration model is a simple and popular way to produce uniform random graphs with a fixed degree sequence. In this talk, we use it to formulate asymptotic conditions on the degree sequence ${d_1, d_2, ..., d_n}$ such that, in the graph generated, the largest component contains $n − o(n)$ vertices. We also provide stricter conditions under which the graph has a non-vanishing probability to be connected. We show that the relevant parameters in this regime are the number of vertices of degree $1$ and $2$, and the average degree, while vertices of higher degree play a minor role. We also derive sharp asymptotics for the number of vertices outside the giant, as a function of these parameters. Our proof uses the method of moments to compute the number of components that are lines or cycles, and employs an exploration algorithm to prove that other kinds of small components are rare. Based on joint work with Remco van der Hofstad and Fiona Sloothaak.

##### Weak* topology and graphons (Jan Grebik), **15:00, D1.07**

I will present how the weak* topology can be used to show that the space of graphons with the cut distance is compact. This is a joint work with Dolezal, Hladky, Rocha and Rozhon.

##### Polynomial Schur's theorem (Hong Liu), 14:00, MS.04

I will discuss the Ramsey problem for ${x,y,z: x+y=p(z)}$ for polynomials $p$ over $\mathbb{Z}$.

##### Solution-free sets of integers (Robert Hancock), **15:00, D1.07**

Given a linear equation $L$, a set $A$ which is a subset of $[n]:={1,...,n}$ is $L$-free if $A$ does not contain any 'non-trivial' solutions to $L$. We consider the following three general questions:

(i) What is the size of the largest $L$-free subset of $[n]$?

(ii) How many $L$-free subsets of $[n]$ are there?

(iii) How many maximal $L$-free subsets of $[n]$ are there?

We determine (i) precisely for several general classes of linear equations $L$ of the form $px+qy=rz$ for fixed positive integers $p, q, r$ where $p\ge q\ge r$. Further, up to a multiplicative constant, we answer (ii) for a wide class of such equations $L$, thereby refining a special case of a result of Green. For all such linear equations $L$, we give an upper bound for (iii). In the case when $p=q\ge 2$ and $r=1$ this bound is exact up to an error term in the exponent. We make use of container and removal lemmas of Green to prove this result. This is joint work with Andrew Treglown.

##### On the rational Turán exponents conjecture (Dong Yeap Kang), 14:00, MS.04

The extremal number ${\rm ex}(n,F)$ of a graph $F$ is the maximum number of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number $r \in [1,2]$ is realisable if there exists a graph $F$ with ${\rm ex}(n , F) = \Theta(n^r)$. Several decades ago, Erdős and Simonovits conjectured that every rational number in $[1,2]$ is realisable. Despite decades of effort, the only known realisable numbers are $1, \frac{7}{5}, 2$, and the numbers of the form $1+\frac{1}{m}$, $2-\frac{1}{m}$, $2-\frac{2}{m}$ for integers $m \geq 1$. In particular, it is not even known whether the set of all realisable numbers contains a single limit point other than two numbers $1$ and $2$.

We discuss some progress on the conjecture of Erdős and Simonovits. First, we show that $2 - \frac{a}{b}$ is realisable for any integers $a,b \geq 1$ with $b>a$ and $b \equiv \pm 1 ~({\rm mod}\:a)$. This includes all previously known ones, and gives infinitely many limit points $2-\frac{1}{m}$ in the set of all realisable numbers as a consequence. Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable. This is joint work with Jaehoon Kim and Hong Liu.

##### Structure of intersecting families and juntas (Andrey Kupavskii), 14:00, MS.04

A family of sets is intersecting if any two sets in the family have non-empty intersection. In this talk, I will summarize some of the recent advances in intersecting families. First, I will discuss one general structure theorem for large intersecting families. Next, I will talk about the use of juntas in the study of intersecting families and how to extend to the range n>Ck the results that were known to be valid for n>n_0(k). Finally, I will discuss sharp junta approximation results for shifted families (that are intersecting but not only).

The last part is based on a joint work with Peter Frankl.

##### Optimal packings of sparse graphs (Peter Allen), 14:00, MS.04

I will describe (another) randomised approach to finding perfect packings of sparse graphs in quasirandom or complete host graphs. The packing algorithm is simple and I will describe it completely; the analysis is not so simple. In particular, we prove that Ringel's Conjecture and the Tree Packing Conjecture are true for almost all trees (or sequences of trees, respectively).

This is joint work with Julia Boettcher, Dennis Clemens, Jan Hladky, Diana Piguet and Anusch Taraz.

##### The largest $K_s$-free induced subgraph in a $K_t$-free graph (Oliver Janzer), 14:00, MS.04

The Erdős-Rogers function measures how large a $K_s$-free induced subgraph there must be in a $K_t$-free graph on n vertices. Good estimates are known when $t=s+1$, but in general there are significant gaps between the best known lower and upper bounds. We have improved the best known upper bound for $s+2\le t\le 2s-1$. In this talk, I shall describe the construction, and sketch the proof that it has the required properties. This is joint work with Tim Gowers.

##### Partial Associativity in Latin Squares (Jason Long), 14:00, MS.04

Latin squares arise from the multiplication tables of groups, but the converse is not true in general. Given a Latin square $A$, we can define a group operation giving $A$ as its multiplication table only when $A$ satisfies a suitable associativity constraint. This observation leads to a natural question, originating with Hrushovski, concerning the '1%' version: if $A$ is only partially associative, can we still obtain something resembling a group structure? I will talk about some joint work with Timothy Gowers on this question.

##### Online Ramsey numbers (David Conlon), 14:00, MS.04

The online Ramsey game is a combinatorial game between two players, Builder and Painter. Starting from an infinite set of isolated vertices, Builder draws an edge at each step and Painter immediately paints the edge red or blue. Builder's goal is to force Painter to create a monochromatic $K_n$ using as few edges as possible. The online Ramsey number $\tilde{r}(n)$ is the minimum number of edges Builder needs to guarantee a win. We discuss recent advances in the study of this function, including an exponential improvement to the lower bound.

Partly based on joint work with Jacob Fox, Andrey Grinshpun and Xiaoyu He.

##### Nonlinear problems in arithmetic Ramsey theory (Sean Prendiville), 14:00, MS.04

Celebrated results of Rado and Szemerédi characterise which systems of linear equations are ‘unbreakable' with respect to finite partitions and passing to a dense subset (respectively). We discuss variants of these results of a nonlinear flavour.

##### The Brown-Erdos-Sos conjecture in groups (Mykhaylo Tyomkyn), 14:00, MS.04

The conjecture of Brown, Erdos and Sos from 1973 states that, for any k \geq 3, if a 3-uniform hypergraph H with n vertices does not contain a set of k+3 vertices spanning at least k edges then it has o(n^2) edges. The case k=3 of this conjecture is the celebrated (6,3)-theorem of Ruzsa and Szemeredi, but for all k \geq 4 the conjecture remains open.

Solymosi suggested to study the Brown-Erdos-Sos conjecture when H consists of triples (a, b, ab) in some finite group \Gamma. In this case he proved that the conjecture holds also for k = 4. We prove it in a strong form for all k, and establish a connection to isoperimetric problems on lattices in the process.

Joint work with R. Nenadov and B. Sudakov

##### Restricted subgraph embeddings (Felix Joos), 14:00, MS.04

In this talk I focus on restricted subgraph embeddings. To be more precise, we allow a host graph $G$ to have an edge colouring and we seek a subgraph $H$ such that every colour appears at most once on the edges of $H$. We only use two simply assumptions. Firstly, every colour appears not too often in $G$ so that it is guaranteed that the number of colours in $G$ is slightly larger than the number of edges of $H$. Secondly, one colour appears not too often at a particular vertex of $G$. Our main result implies in such a setting the existence of $H$ in quasirandom (multipartite) graphs $G$.

This directly implies approximate results of extensions from all trees to all bounded degree graphs of the harmonious labelling conjecture, the Orthogonal Double Cover conjecture, and Kotzig’s conjecture.

This is joint work with Stefan Ehard and Stefan Glock.

##### Judiciously $3$-partitioning $3$-uniform hypergraphs (Marius Tiba), 14:00, MS.04

Bollob\'as, Reed and Thomason proved every $3$-uniform hypergraph $H$ with $m$ edges has a vertex-partition $V=V_1 \sqcup V_2 \sqcup V_3$ such that each part meets at least $\frac{1}{3}(1-\frac{1}{e})m$ edges, later improved to $0.6m$ by Halsegrave and improved asymptotically to $0.65m+o(m)$ by Ma and Yu. We improve this asymptotic bound to $\frac{19}{27}m+o(m)$, which is best possible up to the error term, resolving a special case of a conjecture of Bollob\'as and Scott.

##### (Katherine Staden), 14:00, MS.04

##### Eulerian walks in infinite graphs (Max Pitz), 14:00, MS.04

I explain the development of Eulerian walks in infinite graphs from Erdös et al (1938) via Nash-Williams (1960) to a fairly recent topological solution by Diestel and Kühn (2003), the first solution which restores in the infinite case the intuitive appeal that an Eulerian tour should return again, after visiting all edges, to its start vertex. Then I’ll talk about some new results of myself and co-authors that extend Diestel and Kühn's result to hyperbolic graphs and other graph-like structures.

##### An extremal problem concerning projective cubes (Adam Wagner), 14:00, MS.04

In the Boolean lattice, Sperner's, Erdos's, Kleitman's and Samotij's theorems state that families that do not contain many chains must have a

very specific layered structure. We show that if instead of $Z_2^n$ we work in $Z_{2^n}$, some analogous statements hold if one replaces the word $k$-chain

by projective cube of dimension $2^{k-1}$.

This is joint work with Jason Long.