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$\mathbb{F}_{2}$
For every positive integer d, we show that there must exist an absolute constant 
$c \gt 0$ such that the following holds: for any integer 
$n \geqslant cd^{7}$ and any red-blue colouring of the one-dimensional subspaces of 
$\mathbb{F}_{2}^{n}$, there must exist either a d-dimensional subspace for which all of its one-dimensional subspaces get coloured red or a 2-dimensional subspace for which all of its one-dimensional subspaces get coloured blue. This answers recent questions of Nelson and Nomoto, and confirms that for any even plane binary matroid N, the class of N-free, claw-free binary matroids is polynomially 
$\chi$-bounded.
Our argument will proceed via a reduction to a well-studied additive combinatorics problem, originally posed by Green: given a set 
$A \subset \mathbb{F}_{2}^{n}$ with density 
$\alpha \in [0,1]$, what is the largest subspace that we can find in 
$A+A$? Our main contribution to the story is a new result for this problem in the regime where 
$1/\alpha$ is large with respect to n, which utilises ideas from the recent breakthrough paper of Kelley and Meka on sets of integers without three-term arithmetic progressions.
The accurate quantification of wall-shear stress dynamics is of substantial importance for various applications in fundamental and applied research, spanning areas from human health to aircraft design and optimization. Despite significant progress in experimental measurement techniques and postprocessing algorithms, temporally resolved wall-shear stress fields with adequate spatial resolution and within a suitable spatial domain remain an elusive goal. Furthermore, there is a systematic lack of universal models that can accurately replicate the instantaneous wall-shear stress dynamics in numerical simulations of multiscale systems where direct numerical simulations (DNSs) are prohibitively expensive. To address these gaps, we introduce a deep learning architecture that ingests wall-parallel streamwise velocity fields at 
$y^+ \approx 3.9 \sqrt {Re_\tau }$ of turbulent wall-bounded flows and outputs the corresponding two-dimensional streamwise wall-shear stress fields with identical spatial resolution and domain size. From a physical perspective, our framework acts as a surrogate model encapsulating the various mechanisms through which highly energetic outer-layer flow structures influence the governing wall-shear stress dynamics. The network is trained in a supervised fashion on a unified dataset comprising DNSs of statistically one-dimensional turbulent channel and spatially developing turbulent boundary layer flows at friction Reynolds numbers ranging from 
$390$ to 
$1500$. We demonstrate a zero-shot applicability to experimental velocity fields obtained from particle image velocimetry measurements and verify the physical accuracy of the wall-shear stress estimates with synchronized wall-shear stress measurements using the micro-pillar shear-stress sensor for Reynolds numbers up to 
$2000$. In summary, the presented framework lays the groundwork for extracting inaccessible experimental wall-shear stress information from readily available velocity measurements and thus, facilitates advancements in a variety of experimental applications.
An experimental study was conducted in the CICLoPE long-pipe facility to investigate the correlation between wall-pressure and turbulent velocity fluctuations in the logarithmic region, at high friction Reynolds numbers (
$4794 \lesssim Re_\tau \lesssim 47\,015$). Hereby, we explore the scalability of employing wall-pressure to effectively estimate off-the-wall velocity states (e.g. to be of use in real-time control of wall-turbulence). Coherence spectra for wall-pressure and streamwise (or wall-normal) velocity fluctuations collapse when plotted against 
$\lambda _x/y$ and thus reveals a Reynolds-number-independent scaling with distance-from-the-wall. When the squared wall-pressure fluctuations are considered instead of the linear wall-pressure term, the coherence spectra for the wall-pressure-squared and velocity are higher in amplitude at wavelengths corresponding to large-scale streamwise velocity fluctuations (e.g. at 
$\lambda _x/y = 60$, the coherence value increases from roughly 0.1 up to 0.3). This higher coherence typifies a modulation effect, because low-frequency content is introduced when squaring the wall-pressure time series. Finally, quadratic stochastic estimation is employed to estimate turbulent velocity fluctuations from the wall-pressure time series only. For each 
$Re_\tau$ investigated, the estimated time series and a true temporal measurement of velocity inside the turbulent pipe flow yield a normalised correlation coefficient of 
$\rho \approx 0.6$ for all cases. This suggests that wall-pressure sensing can be employed for meaningful estimation of off-the-wall velocity fluctuations and thus for real-time control of energetic turbulent velocity fluctuations at high-
$Re_\tau$ applications.
In the dynamical systems approach to turbulence, unstable periodic orbits (UPOs) provide valuable insights into system dynamics. Such UPOs are usually found by shooting-based Newton searches, where constructing sufficiently accurate initial guesses is difficult. A common technique for constructing initial guesses involves detecting recurrence events by comparing past and future flow states using their 
$L_2$-distance. An alternative method uses dynamic mode decomposition (DMD) to generate initial guesses based on dominant frequencies identified from a short time series, which are signatures of a nearby UPO. However, DMD struggles with continuous symmetries. To address this drawback, we combine symmetry-reduced DMD (SRDMD) introduced by Marensi et al. (2023, J. Fluid Mech., vol. 954, A10), with sparsity promotion. This combination provides optimal low-dimensional representations of the given time series as a time-periodic function, allowing any time instant along this function to serve as an initial guess for a Newton solver. We also discuss how multi-shooting methods operate on the reconstructed trajectories, and we extend the method to generate initial guesses for travelling waves. We demonstrate SRDMD as a method complementary to recurrent flow analysis by applying it to data obtained by direct numerical simulations of three-dimensional plane Poiseuille flow at the friction Reynolds number 
$\textit{Re}_\tau \approx51$ (
$\textit{Re}=802$), explicitly taking a continuous shift symmetry in the streamwise direction into account. The resulting unstable relative periodic orbits cover relevant regions of the state space, highlighting their potential for describing the flow.
