Semi-algebraic colorings of complete graphs

We consider $m$-colorings of the edges of a complete graph, where each color class is defined semi-algebraically with bounded complexity. The case $m = 2$ was first studied by Alon et al., who applied this framework to obtain surprisingly strong Ramsey-type results for intersection graphs of geometric objects and for other graphs arising in computational geometry. Considering larger values of $m$ is relevant, e.g., to problems concerning the number of distinct distances determined by a point set. For $p\ge 3$ and $m\ge 2$, the classical Ramsey number $R(p;m)$ is the smallest positive integer $n$ such that any $m$-coloring of the edges of $K_n$, the complete graph on $n$ vertices, contains a monochromatic $K_p$. It is a longstanding open problem that goes back to Schur (1916) to decide whether $R(p;m)=2^{O(m)}$, for a fixed $p$. We prove that this is true if each color class is defined semi-algebraically with bounded complexity. The order of magnitude of this bound is tight. Our proof is based on the Cutting Lemma of Chazelle {\em et al.}, and on a Szemer\'edi-type regularity lemma for multicolored semi-algebraic graphs, which is of independent interest. The same technique is used to address the semi-algebraic variant of a more general Ramsey-type problem of Erd\H{o}s and Shelah

Pivot-Point Procedures in Practical Travel Demand Forecasting

For many cities, regions and countries, large-scale model systems have been developed to support the development of transport policy. These models are intended to predict the traffic flows that are likely to result from assumed exogenous developments and transport policies affecting people and businesses in the relevant area. The accuracy of the model is crucial to determining the quality of the information that can be extracted as input to the planning and policy analysis process. A frequent approach to modelling, which can substantially enhance the accuracy of the model, is to formulate the model as predicting changes relative to a base-year situation. Often, base-year traffic flows can be observed rather accurately and the restriction of the model to predicting differences reduces the scope for errors in the modelling – whether they be caused by errors in the model itself or in the inputs to the model – to influence the outputs. Such approaches are called ‘pivot point’ methods, or sometimes incremental models. The approaches have proved themselves beneficial in practical planning situations and now form part of the recommended ‘VaDMA’ (Variable Demand Modelling Advice) guidelines issued by the UK Department for Transport. While the principle of the pivot point is clear, the implementation of the principle in practical model systems can be done in a number of ways and the choice between these can have substantial influence on the model forecasts. In particular modellers need to consider: 1.whether the change predicted by the model should be expressed as an absolute difference or a proportional ratio, or whether a mixed approach is necessary; 2.how to deal with apparently growth in ‘green-field’ situations when applying these approaches; 3,at what level in the model should the pivoting apply, i.e. at the level of mode choice, destination choice, overall travel frequency or combinations of these; 4,whether the pivoting is best undertaken as an operation conducted on a ‘base matrix’ or the model is constructed so that it automatically reproduces the base year situation with base year inputs. The paper reviews the alternative approaches to each of these issues, discussing current practice and attempting to establish the basis on which alternative approaches might be established; in particular, whether pivoting is treated as a correction to a model which is in principle correctly specified but incorporates some error, perhaps from faulty data, or as a partial replacement for a model that handles at best part of the situation. These views of the pivoting lead to different procedures. It goes on to present and justify the approach that the authors have found useful in a number of large-scale modelling studies in The Netherlands, the United Kingdom and elsewhere, pointing out the problems that have led to the calculations that are recommended.

A polynomial regularity lemma for semi-algebraic hypergraphs and its applications in geometry and property testing

Fox, Gromov, Lafforgue, Naor, and Pach proved a regularity lemma for semi-algebraic $k$-uniform hypergraphs of bounded complexity, showing that for each $\epsilon>0$ the vertex set can be equitably partitioned into a bounded number of parts (in terms of $\epsilon$ and the complexity) so that all but an $\epsilon$-fraction of the $k$-tuples of parts are homogeneous. We prove that the number of parts can be taken to be polynomial in $1/\epsilon$. Our improved regularity lemma can be applied to geometric problems and to the following general question on property testing: is it possible to decide, with query complexity polynomial in the reciprocal of the approximation parameter, whether a hypergraph has a given hereditary property? We give an affirmative answer for testing typical hereditary properties for semi-algebraic hypergraphs of bounded complexity

The power of place in disaster recovery: Heritage-based practice in the post-Matthew landscape of Princeville, North Carolina

This article examines shortcomings and possible improvements to standard post-disaster recovery processes through the lens of recovery in Princeville, North Carolina, the oldest black town in the United States. Princeville has faced existential challenges since it was settled in the Tar River floodplain in 1865, most recently in 2016 with flooding caused by Hurricane Matthew. The article describes the power of place attachment and the trauma caused by place-based disaster. It points out that significant rebuilding typically begins a full three years into a standard recovery timeline. And it argues that in the midst of that recovery process, our identification of significant landscapes—i.e., landscapes worth protecting and restoring—is too heavily driven by the object-oriented standards of traditional historic preservation. This article describes work coordinated by North Carolina State University design faculty in partnership with the town of Princeville to supplement abstract, top-down recovery processes with practice that is landscape-based and interactive, that marks histories and establishes concrete symbols of ongoing life, and that promises to help displaced communities to build social-ecological resilience and to heal. This type of work will only become more vital as more communities face climate-induced disasters and the need to rebuild. By describing the impetus and possible impact of NC State’s post-disaster work with Princeville, this article seeks to start a conversation about how our recovery processes can better recognize the power of place and the role of the land as a vehicle for resilience and healing