Abstract
We give an exponential lower bound for the smallest \(N\) such that no matter how we c-color the edges of a complete \(r\)-uniform hypergraph on \(N\) vertices, we can always find a monochromatic Berge-\(K_n\).
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Gerbner and Palmer [5], generalizing the definition of hypergraph cycles due to Berge, introduced the following notion. A hypergraph H contains a Berge copy of a graph G, if there are injections \(\Psi _1: V(G)\rightarrow V(H)\) and \(\Psi _2: E(G) \rightarrow E(H)\) such that for every edge \(uv\in E(G)\) the containment \(\Psi _1(u),\Psi _1(v)\in \Psi _2(uv)\) holds, i.e., each graph edge can be mapped into a distinct hyperedge containing it to create a copy of G. If \(|E(H)|=|E(G)|\), then we say that H is a Berge-G, and we denote such hypergraphs by \({\mathcal {B}}G\).
The study of Ramsey problems for such hypergraphs started independently in 2018 by three groups of authors [1, 4, 6]. Denote by \(R_r({\mathcal {B}}G; c)\) the size of the smallest N such that no matter how we c-color the r-edges of \(K_N^r\), the complete r-uniform hypergraph, we can always find a monochromatic \({\mathcal {B}}G\). In [1] \(R_r({\mathcal {B}}K_n; c)\) was studied for \(n=3,4\). In [4] it was conjectured that \(R_r({\mathcal {B}}K_n; c)\) is bounded by a polynomial of n (depending on r and c), and they showed that \(R_r({\mathcal {B}}K_n; c)=n\) if \(r>2c\) and \(R_r({\mathcal {B}}K_n; c)=n+1\) if \(r=2c\), while \(R_3({\mathcal {B}}K_n; 2)< 2n\) (also proved in [6]). In [6] a superlinear lower bound was shown for \(r=c=3\) and for every other r for large enough c. This was improved in [3] to \(R_{r}({\mathcal {B}}K_n; c)=\Omega (n^{d})\) if \(c>(d-1)\left({{r}\atop {2}}\right)\) and \(R_r({\mathcal {B}}K_n; c)=\Omega (n^{1+1/(r-2)}/\log n)\). We further improve these to disprove the conjecture of [4].
\(R_{r}\left( {\mathcal {B}}K_n; c\right) > \left( 1+\frac{1}{r^2}\right) ^{n-1}\) if \(c>\left({{r}\atop {2}}\right)\).
FormalPara ProofIt is enough to prove the statement for \(c=\left({{r}\atop {2}}\right) +1\). For \(r=2\) this reduces to the classical Ramsey’s theorem, so we can assume \(r\ge 3\). We can also suppose \(n\ge \left({{r}\atop {2}}\right) +1=c\), or the lower bound becomes trivial. Suppose \(N\le (1+\frac{1}{r^2})^{n-1}\). Assign randomly (uniformly and independently) a forbidden color to every pair of vertices in \(K_N^r\). Color the r-edges of \(K_N^r\) arbitrarily, respecting the following rule: if \(\{u,v\}\subset E\), then the color of E cannot be the forbidden color of \(\{u,v\}\). Since \(c>\left({{r}\atop {2}}\right)\), this leaves at least one choice for each edge. Following the classic proof of the lower bound of the Ramsey’s theorem, now we calculate the probability of having a monochromatic \({\mathcal {B}}K_n\). The chance of a monochromatic \({\mathcal {B}}K_n\) on a fixed set of n vertices for a fixed color is at most \((\frac{c-1}{c})^{\left( {{n}\atop{2}}\right)}\), as the fixed color cannot be the forbidden one on any of the pairs of vertices. Thus the expected number of monochromatic \({\mathcal {B}}K_n\)’s is at most \(c\left( {{N}\atop {n}}\right) (\frac{c-1}{c})^{\left( {{n}\atop{2}}\right)}\). If this quantity is less than 1, then we know that a suitable coloring exists. Since \(c\le n\le n!\), it is enough to show that \(N< (\frac{c}{c-1})^{\frac{n-1}{2}}\), but this is true using \(c=\left({{r}\atop {2}}\right) +1\) and \(r\ge 3\). \(\square\)
1 Remarks and Acknowledgment
As was brought to my attention by an anonymous referee, my construction for \(r=3\) and \(c=4\) is essentially the same as the one used in the proof of Theorem 1(ii) in [2] for a different problem, the 4-color Ramsey number of the so-called hedgehog. A hedgehog with body of order n is a 3-uniform hypergraph on \(n+\left( {{n}\atop{2}}\right)\) vertices such that n vertices form its body, and any pair of vertices from its body are contained in exactly one hyperedge, whose third vertex is one of the other \(\left( {{n}\atop{2}}\right)\) vertices, a different one for each hypderedge. It is easy to see that such a hypergraph is a Berge copy of \(K_n\), and while their result, an exponential lower bound for the 4-color Ramsey number of the hedgehog, does not directly imply mine, their construction is such that it also avoids a monochromatic \({\mathcal {B}}K_n\).
It is an interesting problem to determine how \(R_{r}({\mathcal {B}}K_n; c)\) behaves if \(c\le \left({{r}\atop {2}}\right)\). The first open case is \(r=c=3\), just like for hedgehogs.
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Pálvölgyi, D. Exponential Lower Bound for Berge-Ramsey Problems. Graphs and Combinatorics 37, 1433–1435 (2021). https://doi.org/10.1007/s00373-021-02328-3
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DOI: https://doi.org/10.1007/s00373-021-02328-3