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1. Subdivided Claws and the Clique-Stable Set Separation Property

   https://doi.org/10.1007/978-3-030-62497-2  

   Chudnovsky, Maria and (February 2021, MATRIX book series)

   null (Ed.)

   Let C be a class of graphs closed under taking induced subgraphs. We say that C has the clique-stable set separation property if there exists c ∈ N such that for every graph G ∈ C there is a collection P of partitions (X, Y ) of the vertex set of G with |P| ≤ |V (G)| c and with the following property: if K is a clique of G, and S is a stable set of G, and K ∩ S = ∅, then there is (X, Y ) ∈ P with K ⊆ X and S ⊆ Y . In 1991 M. Yannakakis conjectured that the class of all graphs has the clique-stable set separation property, but this conjecture was disproved by M. G¨o¨os in 2014. Therefore it is now of interest to understand for which classes of graphs such a constant c exists. In this paper we define two infinite families S, K of graphs and show that for every S ∈ S and K ∈ K, the class of graphs with no induced subgraph isomorphic to S or K has the clique-stable set separation property. 

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2. Induced subgraphs and tree decompositions III. Three-path-configurations and logarithmic treewidth

   https://doi.org/10.19086/aic.2022.6  

   Chudnovsky, Maria; Abrishami, Tara; Hajebi, Sepehr; Spirkl, Sophie (September 2022, Advances in Combinatorics)

   A _theta_ is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family $$\mathcal{H}$$ of graphs, we say a graph $$G$$ is $$\mathcal{H}$$-_free_ if no induced subgraph of $$G$$ is isomorphic to a member of $$\mathcal{H}$$. We prove a conjecture of Sintiari and Trotignon, that there exists an absolute constant $$c$$ for which every (theta, triangle)-free graph $$G$$ has treewidth at most $$c\log (|V(G)|)$$. A construction by Sintiari and Trotignon shows that this bound is asymptotically best possible, and (theta, triangle)-free graphs comprise the first known hereditary class of graphs with arbitrarily large yet logarithmic treewidth.Our main result is in fact a generalization of the above conjecture, that treewidth is at most logarithmic in $|V(G)|$ for every graph $$G$$ excluding the so-called _three-path-configurations_ as well as a fixed complete graph. It follows that several NP-hard problems such as Stable Set, Vertex Cover, Dominating Set and $$k$$-Coloring (for fixed $$k$$) admit polynomial time algorithms in graphs excluding the three-path-configurations and a fixed complete graph. 

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3. Induced Subgraph Density. I. A loglog Step Towards Erd̋s–Hajnal

   https://doi.org/10.1093/imrn/rnae065  

   Bucić, Matija; Nguyen, Tung; Scott, Alex; Seymour, Paul (May 2024, International Mathematics Research Notices)

   Abstract In 1977, Erd̋s and Hajnal made the conjecture that, for every graph $$H$$, there exists $c>0$ such that every $$H$$-free graph $$G$$ has a clique or stable set of size at least $$|G|^{c}$$, and they proved that this is true with $$ |G|^{c}$$ replaced by $$2^{c\sqrt{\log |G|}}$$. Until now, there has been no improvement on this result (for general $$H$$). We prove a strengthening: that for every graph $$H$$, there exists $c>0$ such that every $$H$$-free graph $$G$$ with $$|G|\ge 2$$ has a clique or stable set of size at least $$ \begin{align*} &2^{c\sqrt{\log |G|\log\log|G|}}.\end{align*} $$ Indeed, we prove the corresponding strengthening of a theorem of Fox and Sudakov, which in turn was a common strengthening of theorems of Rödl, Nikiforov, and the theorem of Erd̋s and Hajnal mentioned above. 

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4. Erdős–Hajnal for graphs with no 5‐hole

   https://doi.org/10.1112/plms.12504  

   Chudnovsky, Maria; Scott, Alex; Seymour, Paul; Spirkl, Sophie (January 2023, Proceedings of the London Mathematical Society)

   Abstract The Erdős–Hajnal conjecture says that for every graph there exists such that every graph not containing as an induced subgraph has a clique or stable set of cardinality at least . We prove that this is true when is a cycle of length five. We also prove several further results: for instance, that if is a cycle and is the complement of a forest, there exists such that every graph containing neither of as an induced subgraph has a clique or stable set of cardinality at least . 

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5. Concatenating Bipartite Graphs

   https://doi.org/10.37236/8451  

   Chudnovsky, Maria; Hompe, Patrick; Scott, Alex; Seymour, Paul; Spirkl, Sophie (April 2022, The Electronic Journal of Combinatorics)

   Let $$x,y\in (0,1]$$, and let $A,B,C$ be disjoint nonempty stable subsets of a graph $$G$$, where every vertex in $$A$$ has at least $x|B|$ neighbours in $$B$$, and every vertex in $$B$$ has at least $y|C|$ neighbours in $$C$$, and there are no edges between $A,C$. We denote by $$\phi(x,y)$$ the maximum $$z$$ such that, in all such graphs $$G$$, there is a vertex $$v\in C$$ that is joined to at least $z|A|$ vertices in $$A$$ by two-edge paths. This function has some interesting properties: we show, for instance, that $$\phi(x,y)=\phi(y,x)$$ for all $x,y$, and there is a discontinuity in $$\phi(x,x)$$ when $1/x$ is an integer. For $z=1/2, 2/3,1/3,3/4,2/5,3/5$, we try to find the (complicated) boundary between the set of pairs $(x,y)$ with $$\phi(x,y)\ge z$$ and the pairs with $$\phi(x,y)1/3$. 

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