On unavoidable‐induced subgraphs in large prime graphs

Abstract

Chudnovsky, Kim, Oum, and Seymour recently established that any prime graph contains one of a short list of induced prime subgraphs [1]. In the present article, we reprove their theorem using many of the same ideas, but with the key model‐theoretic ingredient of first determining the so‐called amount of stability of the graph. This approach changes the applicable Ramsey theorem, improves the bounds and offers a different structural perspective on the graphs in question. Complementing this, we give an infinitary proof that implies the finite result.

Model Theory and Combinatorics of Banned Sequences

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We set up a general context in which one can prove Sauer-Shelah type lemmas. We apply our general results to answer a question of Bhaskar [1] and give a slight improvement to a result of Malliaris and Terry [7]. We also prove a new Sauer-Shelah type lemma in the context of op-rank, a notion of Guingona and Hill [4].

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“…Banned binary sequence problems can be applied to other problems with a tree structure. We use this to improve a result of Malliaris and Terry [7]. Definition 3.12.…”

The stable regularity lemma revisited

“…For the first, inspired by the fact that infinite models of stable theories contain large indiscernible sets, we prove that for finite stable graphs or hypergraphs (indeed, in any finite stable relational structure, suitably defined) one can extract much larger indiscernible sets than expected from Ramsey's theorem, of size n c rather than log n for c depending on the set of relations and their 'stability,' as measured by rank. (This was well exposited, in the case of graphs, in [13], where it found a nice application.) By iteratively using this theorem, one then can, with some additional care, build a first regularity lemma for stable graphs in which all pieces are cliques or independent sets of size n c , plus a remainder, though necessarily the number of pieces grows with the size of the graph.…”

Notes on the Stable Regularity Lemma

“…For the first, inspired by the fact that infinite models of stable theories contain large indiscernible sets, we prove that for finite stable graphs or hypergraphs (indeed, in any finite stable relational structure, suitably defined) one can extract much larger indiscernible sets than expected from Ramsey’s theorem, of size rather than for c depending on the set of relations and their ‘stability’, as measured by rank. (This was well exposited, in the case of graphs, in [14], where it found a nice application.) By iteratively using this theorem, one then can, with some additional care, build a first regularity lemma for stable graphs in which all pieces are cliques or independent sets of size , plus a remainder, though necessarily the number of pieces grows with the size of the graph.…”