Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 1-color-avoiding paths, special tournaments, and incidence geometry   We discuss two approaches to a recent question of Loh: must a 3-colored
transitive tournament on $N$ vertices have a 1-color-\emph{avoiding} path of
vertex-length at least $N^{2/3}$? This question generalizes the
Erd\H{o}s--Szekeres theorem on monotone subsequences.
  First, we define three canonical transformations on these tournaments called
Color, Record, and Dual. We use these to establish a reduction to special
tournaments with natural geometric and combinatorial properties. In many cases
(including all known tight examples), these tournaments have recursive Gallai
decompositions. Not all relevant tournaments have Gallai decompositions, but
those that do satisfy the desired $N^{2/3}$ bound by recent work of Wagner,
roughly analogous to earlier work of Fox, Grinshpun, and Pach on a similar
\emph{undirected} problem.
  Second, we consider the related geometric problem of bounding
\emph{slice-increasing} sets $S\subseteq [n]^3$, which---under an additional
ordering hypothesis on $S$---was shown by Loh to be equivalent to the original
question. In particular, we establish a rigorous connection from a problem of
Szab\'o and Tardos, raise a stronger $L^2$-question on slice-counts, and
mention a surprising overlap with the joints problem.

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The categories of this paper are: ao-sci
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The categories of this paper are: plasm-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: q-alg
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The categories of this paper are: alg-geom
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The categories of this paper are: physics
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The categories of this paper are: stat
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-th
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: comp-gas