Parity of Sets of Mutually Orthogonal Latin Squares

Every Latin square has three attributes that can be even or odd, but any two of these attributes determines the third. Hence the parity of a Latin square has an information content of 2 bits. We extend the definition of parity from Latin squares to sets of mutually orthogonal Latin squares (MOLS) and the corresponding orthogonal arrays (OA). Suppose the parity of an $\mathrm{OA}(k,n)$ has an information content of $\dim(k,n)$ bits. We show that $\dim(k,n) \leq {k \choose 2}-1$. For the case corresponding to projective planes we prove a tighter bound, namely $\dim(n+1,n) \leq {n \choose 2}$ when $n$ is odd and $\dim(n+1,n) \leq {n \choose 2}-1$ when $n$ is even. Using the existence of MOLS with subMOLS, we prove that if $\dim(k,n)={k \choose 2}-1$ then $\dim(k,N) = {k \choose 2}-1$ for all sufficiently large $N$. Let the ensemble of an $\mathrm{OA}$ be the set of Latin squares derived by interpreting any three columns of the OA as a Latin square. We demonstrate many restrictions on the number of Latin squares of each parity that the ensemble of an $\mathrm{OA}(k,n)$ can contain. These restrictions depend on $n\mod4$ and give some insight as to why it is harder to build projective planes of order $n \not= 2\mod4$ than for $n \not= 2\mod4$. For example, we prove that when $n \not= 2\mod 4$ it is impossible to build an $\mathrm{OA}(n+1,n)$ for which all Latin squares in the ensemble are isotopic (equivalent to each other up to permutation of the rows, columns and symbols)

Graph Theory

This workshop focused on recent developments in graph theory. These included in particular recent breakthroughs on nowhere-zero flows in graphs, width parameters, applications of graph sparsity in algorithms, and matroid structure results

The Politics of Progressive Income Taxation with Incentive Effects

This paper studies majority voting over non-linear income taxes when individuals respond to taxation by substituting untaxable leisure to taxable labor (incentive effects). We first show that voting cycle over progressive and regressive taxes is inevitable. This is because the middle-class can always lower its tax burden at the expense of the rich by imposing progressive taxes (convex tax function) while the rich and the poor can reduce their tax burden by imposing regressive taxes (concave tax function). We then investigate three solutions to this cycling problem: (i) reducing the policy space to the policies that are ideal for some voter; (ii) weakening the voting equilibrium concept; (iii) assuming parties also care about the size of their majority. The main results is that progressivity emerges as a voting equilibrium if there is a lack of polarization at the extremes of the income distribution. Interestingly the poor would prefer regressive taxes.Majority voting, Income taxation, Tax progressivity

Generalizing Cross-Document Event Coreference Resolution Across Multiple Corpora

Cross-document event coreference resolution (CDCR) is an NLP task in which mentions of events need to be identified and clustered throughout a collection of documents. CDCR aims to benefit downstream multi-document applications, but despite recent progress on corpora and system development, downstream improvements from applying CDCR have not been shown yet. We make the observation that every CDCR system to date was developed, trained, and tested only on a single respective corpus. This raises strong concerns on their generalizability -- a must-have for downstream applications where the magnitude of domains or event mentions is likely to exceed those found in a curated corpus. To investigate this assumption, we define a uniform evaluation setup involving three CDCR corpora: ECB+, the Gun Violence Corpus and the Football Coreference Corpus (which we reannotate on token level to make our analysis possible). We compare a corpus-independent, feature-based system against a recent neural system developed for ECB+. Whilst being inferior in absolute numbers, the feature-based system shows more consistent performance across all corpora whereas the neural system is hit-and-miss. Via model introspection, we find that the importance of event actions, event time, etc. for resolving coreference in practice varies greatly between the corpora. Additional analysis shows that several systems overfit on the structure of the ECB+ corpus. We conclude with recommendations on how to achieve generally applicable CDCR systems in the future -- the most important being that evaluation on multiple CDCR corpora is strongly necessary. To facilitate future research, we release our dataset, annotation guidelines, and system implementation to the public.Comment: Accepted at CL Journa

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic