A Path Forward: Tropicalization in Extremal Combinatorics
Grigoriy Blekherman, Annie Raymond

TL;DR
This paper explores the structure of binomial inequalities in extremal graph theory, revealing that the set of valid inequalities forms a rational polyhedral cone in several cases, and introduces tropicalization as a key analytical tool.
Contribution
It computes the cone of valid pure binomial inequalities for various graph families, demonstrates polyhedrality, and applies tropicalization to analyze these inequalities.
Findings
The cone of valid inequalities is rational polyhedral for several graph families.
Polyhedrality is proven for series-parallel and chordal graphs.
Tropicalization is established as a useful technique in this context.
Abstract
Many important problems in extremal combinatorics can be be stated as proving a pure binomial inequality in graph homomorphism numbers, i.e., proving that homhomhomhom holds for some fixed graphs and all graphs . One prominent example is Sidorenko's conjecture. For a fixed collection of graphs , the exponent vectors of valid pure binomial inequalities in graphs of form a convex cone. We compute this cone for several families of graphs including complete graphs, even cycles, stars and paths; the latter is the most interesting and intricate case that we compute. In all of these cases, we observe a tantalizing polyhedrality phenomenon: the cone of valid pure binomial inequalities is actually rational polyhedral, and therefore all valid pure…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Combinatorial Mathematics · Graph theory and applications
