A Path Forward: Tropicalization in Extremal Combinatorics

Many important problems in extremal combinatorics can be be stated as proving a pure binomial inequality in graph homomorphism numbers, i.e., proving that hom$(H_1,G)^{a_1}\cdots$hom$(H_k,G)^{a_k}\geq$hom$(H_{k+1},G)^{a_{k+1}}\cdots$hom$(H_m,G)^{a_m}$ holds for some fixed graphs $H_1,\dots,H_m$ and all graphs $G$. One prominent example is Sidorenko's conjecture. For a fixed collection of graphs $\mathcal{U}=\{H_1,\dots,H_m\}$, the exponent vectors of valid pure binomial inequalities in graphs of $\mathcal{U}$ 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 binomial inequalities can be generated from the finite collection of exponent vectors of the extreme rays. Using the work of Kopparty and Rossman, we show that the cone of valid inequalities is indeed rational polyhedral when all graphs $H_i$ are series-parallel and chordal, and we conjecture that polyhedrality holds for any finite collection $\mathcal{U}$. We demonstrate that the polyhedrality phenomenon also occurs in matroids and simplicial complexes. Our description of the inequalities for paths involves a generalization of the Erd\H{o}s-Simonovits conjecture recently proved in its original form by Sa\u{g}lam and a new family of inequalities not observed previously. We also solve an open problem of Kopparty and Rossman on the homomorphism domination exponent of paths. One of our main tools is tropicalization, a well-known technique in complex algebraic geometry. We prove several results about tropicalizations which may be of independent interest.