Discrete Mathematics 2010-2022

Dear colleagues,

we kindly invite you to the {\em Seminar on Analysis and Algebra Alpe-Adria (SAAAA)} on 24th May (Ljubljana).


The SAAAA is a newly established, joint seminar between Zagreb, Ljubljana and Graz in both algebra and analysis, with speakers from all three cities.


Date and location:}
24th May (Saturday), 10:00, at the Faculty of Mathematics and Physics, Jadranska 19, Ljubljana.


Schedule:}
The seminar takes place from 10:00-13:30.\newline
There will be six talks à la 30 minutes (3 talks in analysis and algebra, respectively; two speakers from each city).\newline
Afterwards, a joint lunch in a restaurant is organized.


Speakers:}\newline
Roman Bessonov (University of Ljubljana): TBA\newline
Aleksandar Bulj (University of Zagreb): Powers of unimodular homogeneous multipliers\newline
Nina Kamčev (University of Zagreb): Counting regular hypergraphs\newline
Polona Oblak (University of Ljubljana): TBA\newline
Balint Rago (University of Graz): TBA\newline
Petr Siegl (TU Graz): TBA


Registration:}
If you would like to attend, please register by email (nicolussi@math.tugraz.at).


If you have any further questions, please do not hesitate to ask (nicolussi@math.tugraz.at and frisch@math.tugraz.at).


We are looking forward to seeing you in Ljubljana!


With best wishes,
Sophie Frisch and Noema Nicolussi
(SAAAA organizers at Graz)

Given a prime $p > 3$, we characterize positive-definite integral qua-dratic forms that are coprime-universal for p, i.e. representing all positive integers coprime to p. This generalizes the 290-Theorem by Bhargava and Hanke and extends later works by Rouse (p = 2) and De Benedetto and Rouse (p = 3). When p = 5, 23, 29, 31, our results are conditional on the coprime-universality of specific ternary forms. We prove this assumption under GRH (for Dirichlet and modular L-functions), following a strategy introduced by Ono and Soundararajan, together with some more elementary techniques borrowed from Kaplansky and Bhargava. Finally, we discuss briefly the problem of representing all integers in an arithmetic progression.

Abstract:

We explore two areas of modern statistical analysis. First, we examine overparameterized statistical learning problems, which are central to modern machine learning. We illustrate how results on random matrices can be applied to prove recent breakthrough results concerning the benign effects of overparameterization in a simple setting. Next we turn to statistical challenges in applied optimal transport, with a focus on applications in generative models and barycenters. We will discuss both recent developments and ongoing challenges.
This talk is intended to be accessible to a broad audience, with no specialized background required.

The irrational prime rotation $(p\alpha)_{p \in \mathbb{P}} \mod 1$ is a very classical topic in analytic number theory, going back to the foundational equidistribution result of Vinogradov. In this talk, I will speak about homogeneous and inhomogeneous Diophantine approximation with prime denominator, i.e. we will treat several questions about the set of those $(\alpha,\beta) \in [0,1)^2$ such that for an approximation function $\psi: \mathbb{P} \to [0,1]$,
$ \lVert p\alpha - \beta \rVert < \psi(p)$ holds for infinitely many primes $p$.

Furthermore, I will point out connections to Diophantine approximation with fixed numerator, trace functions and twisted approximation. This talk is partially based on joint work with Emmanuel Kowalski (https://arxiv.org/pdf/2502.08335}).

We will discuss two types of controlled random walks on graphs. In the choice random walk, the controller chooses between two random neighbours at each step. Whereas, in the epsilon-biased random walk the controller instead has a small probability at each step of a free choice of neighbour. We consider the problem of finding optimal strategies for the controller minimising the expected time to hit a given vertex, or visit (cover) all vertices.

We give a potential function argument which allows us to quantify how much the controller can \textquotedblleft boost\textquotedblright the probabilities of rare events using the choice/bias. We will also show how one can use this to prove upper and lower bounds on the speed-up over the simple random walk. We will also discuss our result showing that expansion in a weighted graph is robust to small local changes in edge weights, and how we can apply this to show that the cover time of an expander is $O(n)$, improving on $O(n\log n)$ for the simple random walk.

This is joint work with Agelos Georgakopoulos, John Haslegrave, Sam Olesker-Taylor and Thomas Sauerwald.

Let $T$ be a closed and densely defined operator on a complex Hilbert space, it is well known that the spectrum of the adjoint operator $T^*$ is the complex conjugate of the spectrum of $T$, that is
\begin{equation*}
\sigma(T^*) = \{ \overline{\lambda}\,:\, \lambda \in \sigma(T)\}.
\end{equation*}
In the quaternionic setting, however, one of the most suitable notions of spectrum is the notion of $S$-spectrum, which is related to the invertibility of a second order operator, and this relation is no longer straightforward.
In this talk I will discuss how we can show that the $S$-spectra of $T$ and $T^*$ coincide and we will establish a connection between the $S$-functional calculi of $T$ and $T^*$. I will also mention how this connection plays a crucial role in characterizing the boundedness of the $H^\infty$-functional calculus for sectorial operators.

Perfect graphs can be described as the graphs whose stable set polytopes are defined by their non-negativity and clique inequalities (including edge inequalities). In 1975, Chvátal defined an analogous class of t-perfect graphs, which are the graphs whose stable set polytopes are defined by their non-negativity, edge inequalities, and odd circuit inequalities. We show that t-perfect graphs are 199053-colourable. This is the first finite bound on the chromatic number of t-perfect graphs and answers a question of Shepherd from 1995.

Our proof also shows that every h-perfect graph with clique number $\omega$ is $(\omega + 199050)$-colourable.

This is joint work with Maria Chudnovsky, Linda Cook, James Davies, and Jane Tan.

A Lipschitz function on a graph $G$ is a function $f : V \rightarrow Z$ from the vertex set of the graph to the integers which changes by at most 1 along any edge of the graph. Given a finite connected graph $G$, and fixing the value of the function to be $0$ on at least one vertex, we may sample such a Lipschitz function uniformly at random. What can we say about the typical height at a vertex? This depends heavily on $G$. For example, when $G$ is a path of length $n$, and the height at one of the endpoints is fixed to be $0$, this model corresponds to a simple random walk with uniform increments in $\{-1,0,1\}$, and hence the height at the opposite endpoint of the path is typically of order $\sqrt{n}$. In this talk, we consider the case when $G$ is a finite tree (to be thought of an infinite tree pruned at a given depth), and the height at the leaves is fixed to be $0$. We would like to understand the distribution of the height at the root. We first discuss the issue of declocalization vs localization (i.e., tightness of the height as the depth of the tree increases), showing that the height function is localized on any transient tree. We then discuss the question of whether the height at the root converges in distribution on a $d$-ary tree, showing that there is a phase transition in $d$.

Based on joint works with Nathaniel Butler, Alon Heller, Kesav Krishnan and Gourab Ray.

Meeting link:

https://tugraz.webex.com/tugraz/j.php?MTID=me6334298e9cfe1dc540f9578342d5308

Motivated by the study of the propagation of convexity by semi-groups of stochastic differential equations and convex comparison between the distributions of solutions of two such equations, we study the comparison for the convex order between a Gaussian distribution and a Gaussian mixture. We give and discuss intrinsic necessary and sufficient conditions for convex ordering.

Consider a stationary, weakly dependent sequence of random variables. Given only mild conditions, allowing for polynomial decay of the autocovariance function, we show a Berry-Esseen bound of optimal order $n^{-1/2}$ for studentized (self-normalized) partial sums, both for the Kolmogorov and Wasserstein (and $L^p$) distance. The results show that, in general, (minimax) optimal estimators of the long-run variance lead to suboptimal bounds in the central limit theorem, that is, the rate $n^{-1/2}$ cannot be reached. This can be salvaged by simple methods: In order to maintain the optimal speed of convergence $n^{-1/2}$, simple over-smoothing within a certain range is necessary and sufficient. The setup contains many prominent dynamical systems and time series models, including random walks on the general linear group, products of positive random matrices, functionals of Garch models of any order, functionals of dynamical systems arising from SDEs, iterated random functions and many more.

Paper: https://ems.press/journals/jems/articles/14298460

Given a bipartite, regular graph $G$ with certain expansion properties, we explore the partition function of the independence polynomial of $G$. If the graph $G$ is deterministic, this leads to studying the hard-core model on independent sets. Using the cluster expansion method, one can obtain arbitrarily precise asymptotics for the number of independent sets. Once the graph $G_p$ is percolated, i.e., each edge is present with probability $p$, independently, we explore a connection to the Ising model. Using this, we are able to extend the work on deterministic graphs and provide an expansion of the expected number of independent sets in $G_p$.

Recently, motivated by applications in spatial biology, we explored the interactions between color classes in a colored point set from a topological perspective. We introduced MST-ratio as a measure for quantifying the mingling of points with different colors. Investigating this measure raises intriguing questions in discrete geometry, which is the primary focus of this talk.
In this talk, I will introduce the concept of the MST-ratio, present the best-known bounds and key complexity results for computing its maximum, and share new findings on its behavior in both random and arbitrary point sets. Finally, I will highlight several open questions in discrete and stochastic geometry that arise from this work.

We consider sequences of finite weighted random graphs that converge locally to unimodular i.i.d. weighted random trees. Examples include sparse Erdos-Renyi random graphs and configuration models. When the weights are atomless, we prove that the matchings of maximal weight converge locally to a matching on the limiting tree. For this purpose, we introduce and study unimodular matchings on weighted unimodular random trees as well as a notion of optimality for these objects. In this context, we prove that, in law, there is a unique optimal unimodular matching for a given unimodular tree. We then prove that this law is the local limit of the sequence of matchings of maximal weight. Along the way, we also show that this law is characterised by an equation derived from a message passing algorithm.

Based on joint works with Nathanael Enriquez, Mike Liu and Vianney Perchet.

Meeting link:

https://tugraz.webex.com/tugraz/j.php?MTID=m01b0553e547155cca576e9d6e12f2c55

It is an associated event of the FWF SFB (F1002) \textquotedblleft Discrete random structures: enumeration and scaling limits\textquotedblright (https://sfbrandom.univie.ac.at/).}

Recently, Ballantine and Welch considered two classes of integer partitions which they labeled POND and PEND partitions. These are integer partitions wherein the odd parts (respectively, the even
parts) *cannot* be distinct. In recent work, I studied these two types of partitions from an arithmetic perspective and proved infinite families of mod 3 congruences satisfied by the two corresponding enumerating functions. I will talk about the generating functions for these enumerating functions, and I will also highlight the elementary proofs that I utilized.

In the latter portion of the talk, I will discuss unexpected connections between these divisibility properties for POND and PEND partitions by considering modified versions of the generating functions in question and relating these new generating functions in a natural way via Atkin-Lehner involutions. This part of the talk is joint work with Nicolas Smoot (University of Vienna).

We introduce a novel method for data-driven tuning of regularization parameters in total-variation image denoising. Our approach leverages the semi-dual Brenier formulation of weak optimal transport between the distributions of clean and noisy images to devise a new loss function for total variation parameter learning. Our loss has a close connection to the traditional bilevel quadratic setting, but it leads to fully explicit monolevel problems, which are, in fact, convex under certain conditions. For training, we introduce a new conditional-gradient-type method, which can handle a complex and unbounded constraint set with computations up to numerical precision. Numerical experiments demonstrate the effectiveness of our approach and suggest promising avenues for future extensions.

For the damped wave equation on the torus, the energy decay rate is known to depend on the geometry of the support of the damping, and growth properties of the damping near where it is zero. In prior work, these growth properties are polynomial bounds on the damping, or derivative bound conditions, which control the gradient of the damping by a power of the damping. In this talk, we will show how these rates can be improved, and generalized to exponentially, or poly-logarithmically, growing damping, as well as more general non-polynomial derivative bound conditions. The proof of these results relies on resolvent estimates on very fine semiclassical scales. Time permitting, we will discuss how these new decay rates change when the geometry of the support of the damping changes.

ABSTRACT:

The talk is about the barycenters of $N$ probability measures with respect to the p-Wasserstein metric ($p>1$), which generalizes the notion of Wasserstein barycenters for $p=2$, introduced by Agueh and Carlier.
Providing a natural way to interpolate probability measures and computing a representative summary of input datasets, they are useful tools in data science, statistics, and image processing.
This is a highly nonlinear problem but it can be rewritten as an equivalent multi-marginal optimal transport problem, paying with an (a priori) increase in dimension.

Here we show that thanks to a new technique based on the geometric properties of the support of the optimal plan, the $p$-Wasserstein barycenters of absolutely continuous marginals are unique and absolutely continuous. This implies that the optimal MMOT plan is unique and can be parametrized as a graph over any marginal space (with a consequent dimension reduction).
Some examples in one dimension are also discussed, with emphasis on the statistical meaning of the $p$-Wasserstein barycenters and on the two natural limits $p\to1$ and $p\to\infty$.

This is a joint work with G. Friesecke and T. Ried.

For $n > 1$ the category of $n$-dim. persistence modules has wild
representation type, that is, there is no simple description of
indecomposable objects. Therefore, the well-known barcode invariant of
1-dim. persistence modules fails to extend to higher dimensions. An
alternative for higher dimensions are the flat-injective ({\it flange})
presentations, which were introduced by Miller as a homological invariant
of persistence modules. We give a criterion for minimality of
flat-injective presentations over local graded rings, and further provide
a construction procedure for finitely supported ${\mathbb Z}^n$-graded modules.
Furthermore, we also discuss minimality of related invariants from the
perspective of relative homological algebra.

Abstract:

The orientations of symmetrical objects in three-dimensional Euclidean space
correspond to elements of the quotient space SO(3)/K, where K denotes the symmetry group.
Such objects naturally occur in crystallography, materials science, biochemistry, and other fields.
When performing the statistical analysis of these manifold-valued data, it is essential to
account for the specific properties of the space SO(3)/K. In this talk, we will discuss selected
statistical problems, with a particular focus on independence testing.

Dear Colleagues,

in the coming semester our Guest Professor J. Royer (Toulouse) will give the following lecture within MAT.637UF Elective subject mathematics (Analysis), see details below.

Title: An Introduction to Semiclassical Analysis

When/where: Tue and Wed 16-18 in STEG006 (Steyrergasse 30)
(start 4.3., details to be confirmed during the first week)

Contents: The main objective of the course is to give an introduction to semiclassical analysis. Semiclassical analysis makes in particular the link between quantum mechanics and classical mechanics, but more generally it shows up naturally in various settings when studying the asymptotic behavior for the solution of a PDE depending on a parameter.

An important part of the course will be devoted to the definition and general properties of pseudo-differential operators (quantizations, composition, action on $L^2$, Egorov$^\prime$s Theorem, G\aa rding Inequality, etc.).

Then we will see how we can use semiclassical tools to (micro)localize the solutions of some PDEs. The main application we will have in mind will be the damped wave equation. In particular, we will see how the famous Geometric Control Condition plays a crucial role for the contribution of high frequencies.

We study some spectral properties of generalized MIT bag models. These are a family of Dirac operators acting on domains of $\mathbb R^3$, $\{\mathcal H_\tau\}_{\tau \in \mathbb R}$. They are used in the field of relativistic quantum mechanics to model confinement of quarks in hadrons, and their energies are related with the spectra of such operators. Their lowest positive eigenvalue is of special interest, and it is conjectured to be minimal for a ball among all domains of the same volume. The analogous conjecture holds true for the Dirichlet Laplacian (it is the Faber-Krahn inequality), which arises in the limit $\tau \to \pm \infty$. Studying the resolvent convergence of $\mathcal H_\tau$ in this limit, some spectral properties of the limiting operators $\mathcal H_{\pm \infty}$ are inherited throughout the parameterization.

Consider the following experiment: a non-relativistic quantum particle is initially prepared with wave function supported in a bounded region $\Omega$ of physical space, and suppose detectors are placed along the boundary of this region. We allow the wave function to evolve in $\Omega$ until the particle is detected along the boundary, at which point we record the time and position of detection. If we perform this experiment repeatedly, the Born rule predicts a distribution for the detected positions of the particle. However, a theoretical description for the distribution of detection times has remained an open problem in quantum mechanics. In this talk we present a recent proposal of Tumulka for the detection time distribution and apply the theory of boundary tuples to prove his model is singled out by a short list of physical assumptions.

Stochastic modeling is a powerful tool in creating so-called digital
twins of micro- and nanostructures of different functional materials.
Here, a digital twin is a virtual sample that is statistically similar
to the original physical twin in its appearance and functional
properties. We present this approach at the example of two different materials.
First, the morphology of nanoporous glass is modeled by use
of an excursion set of a Gaussian random field. Second, filter cakes resulting form a filtration process are modeled by virtually generating individual particles using a mixed Gaussian random field on a sphere and
spatially packing them into an artificial filter cake.

We will discuss Diophantine approximations of real numbers by rationals. We will define the irrationality measure function and explore its relation to best approximations, badly approximable numbers, and the Dirichlet spectrum. A surprising result by Kan and Moshchevitin shows that any two irrationality measure functions interchange infinitely often. We will present a generalization of this result to the case of more than two functions. Finally, we will discuss a lower bound on the difference between these functions.

It is known that many arithmetic functions such as the Chebyshev $\psi$-function or the Mertens (summatory Möbius) function possess well-defined limiting distributions. Given this, it is natural to ask for functional versions of these results, i.e. whether these functions possess limiting stochastic processes. In this talk, we shall discuss a class of stationary stochastic processes with discrete spectrum which appear as functional limits of arithmetic functions. The talk is based on a joint work with A. Iksanov and A. Marynych.

13:30 - 14:30: Sam Chow (University of Warwick)}

Smooth discrepancy and Littlewood’s conjecture

We establish a deterministic analogue of Beck’s local-to-global principle for Kronecker sequences. This gives rise to a novel reformulation of Littlewood’s conjecture in diophantine approximation.

15:00 - 16:00: Matteo Verzobio (ISTA)}

On the L-polynomials of curves over finite fields

Let $q$ be a prime power and $g\geq 1$. Consider a smooth projective curve $C$ of genus $g$ defined over the finite field $\mathbb{F}_q$​. We introduce the L-polynomial associated with $C$. Additionally, we examine the distribution of the coefficients of L-polynomials in the family of curves with genus $g$.

16:15 - 17:15: Andrei Shubin (TU Graz)}

Prime number theorem for sums of digits in several bases

In 1967, Gelfond established an asymptotic formula for the sum of digits of an integer $n$ in base $q$ in arithmetic progressions. He posted a few questions about the distribution of sums of digits along subsequences, such primes and integer polynomials. The formula for primes was established by Mauduit and Rivat, and later Drmota, Mauduit, and Rivat extended this result to two different bases simultaneously.

I will talk about the proof for any number of bases. This is a joint work with Clemens Müllner and Lukas Spiegelhofer.


Seminar webpage with further information:

https://sites.google.com/view/gntd/start

A classical question in extremal (hyper)graph theory asks for tight minimum degree conditions which force the existence of certain spanning structures in large graphs, generalising Dirac's theorem from 1952. One aspect of this concerns tiling graphs with identical vertex disjoint copies of a small subgraph. For example, asking for tight minimum codegree conditions in a $k$-uniform hypergraph which force a perfect matching (under the obvious additional necessary condition that the number of vertices is divisible by $k$). Whilst there has been a lot of interest in these types of tiling problems, still very few results are known. We share a new result in this area, which is joint work with Amarja Kathapurkar, Natasha Morrison and Richard Mycroft.

In this talk, I will present recent results on Schrödinger operators with transmission conditions along a Lipschitz (non-smooth) curves, extending previous work by Behrndt, Holzmann, and Stenzel on smooth curves. Given bounded domain $\Omega_+\subset\mathbb{R}^2$ with a Lipschitz boundary $\Sigma\subset\mathbb{R}^2$ and a unit normal $N=(n_1,n_2)$, and define its complementary exterior domain $\Omega_-$. For a parameter $\alpha\in \mathbb{R}$, the Schrödinger operator we consider acts as
\[
(u_+,u_-)\mapsto (-\Delta u_+,- \Delta u_-) \quad\text{for }\, (u_+,u_-)\in L^2(\Omega_+)\oplus L^2(\Omega_-) \simeq L^2(\mathbb{R}^2),\]
with the so-called oblique transmission condition}:
\begin{equation*}
(n_1+i n_2)(\gamma^+f_+-\gamma^-f_- ) +\alpha(\gamma^+\partial_{\bar{z}} f_+ +\gamma^-\partial_{\bar{z}} f_-)=0 \text{ on } \Sigma.
\end{equation*}



Earlier works focused on smooth curves, showing that these operators are self-adjoint, with well-understood spectral properties. In particular, the discrete spectrum is empty for $\alpha\ge 0$ and infinite and unbounded from below for $\alpha<0$, without accumulation at $0$, and for any fixed $n\in\mathbb{N}$ the $n$-th discrete eigenvalue $\lambda_n$ (if counted with multiplicities in the non-increasing order) satisfies
\begin{equation}
\label{enta}
\lambda_n=-\frac{4}{\alpha^2}+O(1) \text{ for } \alpha\to 0^-.
\end{equation}
Our work generalizes these results to the case where the curve is only Lipschitz continuous. I will show that, while the operator remains self-adjoint and retains some of its spectral structure, the non-smoothness of the boundary introduces significant changes and the asymptotic \eqref{enta} turns out to be false in general. Namely, using a relation with $\delta$-potentials, we will show the two-sided estimate: for any $n\in\mathbb{N}$ there are $0<A<B$ such that
\[
-\frac{B}{\alpha^2}\le
\lambda_n\le-\frac{A}{\alpha^2} \text{ for }\alpha\to 0^-.
\]
Then, we will show that for any $B\in(1,4)$ there exists a non-smooth curve $\Sigma$ such that for the associated operator there holds
\[
\lambda_1=-\dfrac{B}{\alpha^2}+o\Big(\frac{1}{\alpha^2}\Big) \text{ for } \alpha\to 0^-,
\]
which is clearly different from \eqref{enta}.

Based on joint work with Miguel Camarasa (BCAM, Bilbao) and Konstantin Konstantin (University of Oldenburg).

DER VORTRAG WIRD AUF ANFANG DES SOMMERSEMESTERS VERSCHOBEN!

The general framework for handling symbolic summation, namely $\Pi\Sigma$-fields was introduced by Michael Karr. He developed algorithmically how indefinite nested sums and products can be represented as transcendental extensions over a computable ground field \mathbb{K}. He also presented an algorithm that solves the parameterized telescoping problem, and as special case the telescoping and creative telescoping problem within a given $\Pi\Sigma$-field. In recent years, Karr’s difference field theory has been extended by Carsten Schneider to the so-called $\mathrm{R}\Pi\Sigma^{*}$-extensions in which one can represent not only indefinite nested sums and products that can be expressed by transcendental ring extensions, but algebraic products of the form $\alpha^{n}$ where $\alpha$ is a primitive root of unity can also be treated.

In this talk, I will demonstrate how products over primitive roots of unity can be modelled in the so called $\mathrm{R}$-ring extension of a given computable algebraic field $\mathrm{K}$. As a consequence, the new algorithmically constructed ring extension is not an integral domain, i.e., it has zero divisors, which can be recognised.

Combinatorial rigidity theory addresses questions such as: given a structure defined by geometric constraints, what can be inferred about its geometric behaviour based solely on its underlying combinatorial data? Such structures are often modelled as assemblies of rigid rods connected by rotational joints, in which case the underlying combinatorial data is a graph. A typical question in this context is: given such a framework in generic position in $R^d$, is it rigid? That is, does every continuous motion of the vertices (joints) that preserves the lengths of all edges (rods) necessarily preserve the distances between all pairs of vertices?

In this talk, I will present new sufficient conditions for the rigidity of a framework in $R^d$ based on the notion of rigid partitions - partitions of the underlying graph that satisfy certain connectivity properties. I will outline several broadly applicable conditions for the existence of such partitions and discuss a few applications, including results on the rigidity of (pseudo)random graphs.

If time allows, I will also discuss new - often sharp - sufficient minimum degree conditions for $d$-dimensional rigidity and mention a related novel result on the pseudoachromatic number of graphs.

The talk is based on joint work with Michael Krivelevich and Alan Lew.

Meeting link:

https://tugraz.webex.com/tugraz/j.php?MTID=m01b0553e547155cca576e9d6e12f2c55

The vertex-Turan problem for hypercubes asks: how small a family of vertices $F$ can we take in $\{0,1\}^n$, in such a way that $F$ intersects the vertex-set of every $d$-dimensional subcube? A widely-believed folklore conjecture stated that the minimal measure of such a family is (asymptotically) $1/(d+1)$, which is attained by taking every $(d+1)$th layer of the cube. (This was proven in the special case $d=2$ by Kostochka in 1976, and independently by Johnson and Entringer.) In this talk, we will outline a construction of such a family F with measure at most $c^d$ for an absolute constant $c<1$, disproving the folklore conjecture in a strong sense. We will explain the connection to Turan questions for {\lq}daisies{\rq}, and discuss various other widely-believed conjectures, e.g. on forbidden posets, that can be seen to fail due to our construction. Several open problems remain, including the optimal value of $c$ above. Based on joint work with Maria-Romina Ivan and Imre Leader.