Exponential Lower Bound for Berge-Ramsey Problems

We give an exponential lower bound for the smallest N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document} such that no matter how we c-color the edges of a complete r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r$$\end{document}-uniform hypergraph on N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document} vertices, we can always find a monochromatic Berge-Kn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_n$$\end{document}.

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 W 1 : VðGÞ ! VðHÞ and W 2 : EðGÞ ! EðHÞ such that for every edge uv 2 EðGÞ the containment W 1 ðuÞ; W 1 ðvÞ 2 W 2 ðuvÞ holds, i.e., each graph edge can be mapped into a distinct hyperedge containing it to create a copy of G. If jEðHÞj ¼ jEðGÞj, then we say that H is a Berge-G, and we denote such hypergraphs by BG.
The study of Ramsey problems for such hypergraphs started independently in 2018 by three groups of authors [1,4,6]. Denote by R r ðBG; cÞ the size of the smallest N such that no matter how we c-color the r-edges of K r N , the complete runiform hypergraph, we can always find a monochromatic BG. In [1] R r ðBK n ; cÞ was studied for n ¼ 3; 4. In [4] it was conjectured that R r ðBK n ; cÞ is bounded by a polynomial of n (depending on r and c), and they showed that R r ðBK n ; cÞ ¼ n if r [ 2c and R r ðBK n ; cÞ ¼ n þ 1 if r ¼ 2c, while R 3 ðBK n ; 2Þ\2n (also proved in [6]). In [6] a superlinear lower bound was shown for r ¼ c ¼ 3 and for every other r for large enough c. This was improved in [3] and R r ðBK n ; cÞ ¼ Xðn 1þ1=ðrÀ2Þ = log nÞ. We further improve these to disprove the conjecture of [4].
Proof It is enough to prove the statement for c ¼ r 2 À Á þ 1. For r ¼ 2 this reduces to the classical Ramsey's theorem, so we can assume r ! 3. We can also suppose n ! r 2 À Á þ 1 ¼ c, or the lower bound becomes trivial. Suppose N ð1 þ 1 r 2 Þ nÀ1 . Assign randomly (uniformly and independently) a forbidden color to every pair of vertices in K r N . Color the r-edges of K r N arbitrarily, respecting the following rule: if fu; vg & E, then the color of E cannot be the forbidden color of fu; vg. Since c [ r 2 À Á , 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 BK n . The chance of a monochromatic BK n on a fixed set of n vertices for a fixed color is at most ð cÀ1 c Þ ð Þ . If this quantity is less than 1, then we know that a suitable coloring exists. Since c n n!, it is enough to show that N\ð c cÀ1 Þ nÀ1 2 , but this is true using c ¼ r 2 À Á þ 1 and r ! 3. h

Remarks and Acknowledgment
As was brought to my attention by an anonymous referee, my construction for r ¼ 3 and c ¼ 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 þ n 2 À Á 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 n 2 À Á 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 BK n .
It is an interesting problem to determine how R r ðBK n ; cÞ behaves if c r 2 À Á . The first open case is r ¼ c ¼ 3, just like for hedgehogs.
Funding Open access funding provided by Eötvös Loránd University.
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