Beyond chromatic threshold via the (p,q)-theorem, and a sharp blow-up phenomenon
International Symposium on Computational Geometry(2024)
摘要
We establish a novel connection between the well-known chromatic threshold
problem in extremal combinatorics and the celebrated (p,q)-theorem in
discrete geometry. In particular, for a graph G with bounded clique number
and a natural density condition, we prove a (p,q)-theorem for an abstract
convexity space associated with G. Our result strengthens those of Thomassen
and Nikiforov on the chromatic threshold of cliques. Our (p,q)-theorem can
also be viewed as a χ-boundedness result for (what we call) ultra maximal
K_r-free graphs.
We further show that the graphs under study are blow-ups of constant size
graphs, improving a result of Oberkampf and Schacht on homomorphism threshold
of cliques. Our result unravels the cause underpinning such a blow-up
phenomenon, differentiating the chromatic and homomorphism threshold problems
for cliques. It implies that for the homomorphism threshold problem, rather
than the minimum degree condition usually considered in the literature, the
decisive factor is a clique density condition on co-neighborhoods of vertices.
More precisely, we show that if an n-vertex K_r-free graph G satisfies
that the common neighborhood of every pair of non-adjacent vertices induces a
subgraph with K_r-2-density at least ε>0, then G must be a
blow-up of some K_r-free graph F on at most
2^O(r/εlog1/ε) vertices. Furthermore,
this single exponential bound is optimal. We construct examples with no
K_r-free homomorphic image of size smaller than
2^Ω_r(1/ε).
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