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Let $$\phi(x,y)$$ be a continuous function, smooth away from the diagonal, such that, for some $$\alpha>0$$, the associated generalized Radon transforms \begin{equation} \label{Radon} R_t^{\phi}f(x)=\int_{\phi(x,y)=t} f(y) \psi(y) d\sigma_{x,t}(y) \end{equation} map $$L^2({\mathbb R}^d) \to H^{\alpha}({\mathbb R}^d)$$ for all $t>0$. Let $$E$$ be a compact subset of $${\mathbb R}^d$$ for some $$d \ge 2$$, and suppose that the Hausdorff dimension of $$E$$ is $$>d-\alpha$$. We show that any tree graph $$T$$ on $k+1$ ($$k \ge 1$$) vertices is realizable in $$E$$, in the sense that there exist distinct $$x^1, x^2, \dots, x^{k+1} \in E$$ and $t>0$ such that the $$\phi$$-distance $$\phi(x^i, x^j)$$ is equal to $$t$$ for all pairs $(i,j)$ corresponding to the edges of the graph $$T$$.
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On Ramsey numbers of hedgehogs
Abstract The hedgehog H t is a 3-uniform hypergraph on vertices $$1, \ldots ,t + \left({\matrix{t \cr 2}}\right)$$ such that, for any pair ( i , j ) with 1 ≤ i < j ≤ t , there exists a unique vertex k > t such that { i , j , k } is an edge. Conlon, Fox and Rödl proved that the two-colour Ramsey number of the hedgehog grows polynomially in the number of its vertices, while the four-colour Ramsey number grows exponentially in the square root of the number of vertices. They asked whether the two-colour Ramsey number of the hedgehog H t is nearly linear in the number of its vertices. We answer this question affirmatively, proving that r ( H t ) = O ( t 2 ln t ).
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- Award ID(s):
- 1855635
- PAR ID:
- 10178082
- Date Published:
- Journal Name:
- Combinatorics, Probability and Computing
- Volume:
- 29
- Issue:
- 1
- ISSN:
- 0963-5483
- Page Range / eLocation ID:
- 101 to 112
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/s0963548319000312
