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Abstract In this paper, we are interested in the following question: given an arbitrary Steiner triple systemonvertices and any 3‐uniform hypertreeonvertices, is it necessary thatcontainsas a subgraph provided? We show the answer is positive for a class of hypertrees and conjecture that the answer is always positive.
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In this paper we make a partial progress on the following conjecture: for every $$\mu>0$$ and large enough $$n$$, every Steiner triple system $$S$$ on at least $$(1+\mu)n$$ vertices contains every hypertree $$T$$ on $$n$$ vertices. We prove that the conjecture holds if $$T$$ is a perfect $$d$$-ary hypertree.more » « less
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Abstract Let $$\mathrm{AP}_k=\{a,a+d,\ldots,a+(k-1)d\}$$ be an arithmetic progression. For $$\varepsilon>0$$ we call a set $$\mathrm{AP}_k(\varepsilon)=\{x_0,\ldots,x_{k-1}\}$$ an $$\varepsilon$$ -approximate arithmetic progression if for some a and d , $$|x_i-(a+id)|<\varepsilon d$$ holds for all $$i\in\{0,1\ldots,k-1\}$$ . Complementing earlier results of Dumitrescu (2011, J. Comput. Geom. 2 (1) 16–29), in this paper we study numerical aspects of Van der Waerden, Szemerédi and Furstenberg–Katznelson like results in which arithmetic progressions and their higher dimensional extensions are replaced by their $$\varepsilon$$ -approximation.more » « less
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