The size‐Ramsey number of powers of paths

Given graphs G and H and a positive integer q , say that G is q ‐Ramsey for H , denoted G→(H)q , if every q ‐coloring of the edges of G contains a monochromatic copy of H . The size‐Ramsey number rˆ(H) of a graph H is defined to be rˆ(H)=min{∣E(G)∣:G→(H)2} . Answering a question of Conlon, we prove that, for every fixed k , we have rˆ(Pnk)=O(n) , where Pnk is the k th power of the n ‐vertex path Pn (ie, the graph with vertex set V(Pn) and all edges {u,v} such that the distance between u and v in Pn is at most k ). Our proof is probabilistic, but can also be made constructive.

[1]  Yoshiharu Kohayakawa,et al.  The size Ramsey number of short subdivisions of bounded degree graphs , 2018, Random Struct. Algorithms.

[2]  Chunlin You,et al.  A note on the size Ramsey number of powers of paths , 2018 .

[3]  Julia Böttcher.  Large-scale structures in random graphs , 2017, BCC.

[4]  Andrzej Dudek,et al.  On the Size‐Ramsey Number of Hypergraphs , 2015, J. Graph Theory.

[5]  Alexey Pokrovskiy.  Calculating Ramsey Numbers by Partitioning Colored Graphs , 2017, J. Graph Theory.

[6]  David Conlon,et al.  Recent developments in graph Ramsey theory , 2015, Surveys in Combinatorics.

[7]  Shoham Letzter.  Path Ramsey Number for Random Graphs , 2016, Comb. Probab. Comput..

[8]  Andrzej Dudek,et al.  An Alternative Proof of the Linearity of the Size-Ramsey Number of Paths , 2015, Comb. Probab. Comput..

[9]  Alexey Pokrovskiy.  Partitioning edge-coloured complete graphs into monochromatic cycles and paths , 2014, J. Comb. Theory, Ser. B.

[10]  Graham R. Brightwell,et al.  Ramsey-goodness—and otherwise , 2010, Comb..

[11]  Domingos Dellamonica.  The size-Ramsey number of trees , 2012, Random Struct. Algorithms.

[12]  B. Sudakov,et al.  The size Ramsey number of a directed path , 2010, J. Comb. Theory, Ser. B.

[13]  Yoshiharu Kohayakawa,et al.  Sparse partition universal graphs for graphs of bounded degree , 2011 .

[14]  David Reimer,et al.  The Ramsey size number of dipaths , 2002, Discret. Math..

[15]  Yoshiharu Kohayakawa,et al.  The size-Ramsey number of trees , 1995 .

[16]  Yoshiharu Kohayakawa,et al.  The Induced Size-Ramsey Number of Cycles , 1995, Combinatorics, Probability and Computing.

[17]  Xin Ke.  The Size Ramsey Number of Trees with Bounded Degree , 1993, Random Struct. Algorithms.

[18]  Noga Alon,et al.  Explicit construction of linear sized tolerant networks , 1988, Discret. Math..

[19]  Joel Friedman,et al.  Expanding graphs contain all small trees , 1987, Comb..

[20]  József Beck,et al.  On size Ramsey number of paths, trees, and circuits. I , 1983, J. Graph Theory.

[21]  Paul Erdös,et al.  On the combinatorial problems which I would most like to see solved , 1981, Comb..

[22]  P. Erdos,et al.  The size Ramsey number , 1978 .

[23]  V. Sós,et al.  On a problem of K. Zarankiewicz , 1954 .

[24]  Frank Plumpton Ramsey,et al.  On a Problem of Formal Logic , 1930 .

[25]  P. Erdos-L Lovász.  Problems and Results on 3-chromatic Hypergraphs and Some Related Questions , 2022 .