The Existence Spectrum for Overlarge Sets of Pure Hybrid Triple Systems

Abstract

An overlarge set of pure Hybrid triple system $$(PHTS)$$(PHTS), denoted by $$OLPHTS(v)$$OLPHTS(v), is a collection $$\{(Y{\setminus }\{y_i\},{\mathcal {A}}_i)\}_i$${(Y\{yi},Ai)}i, where $$Y$$Y is a $$(v+1)$$(v+1)-set, $$y_i\in Y$$yi?Y, each $$(Y{\setminus }\{y_i\},{\mathcal {A}}_i)$$(Y\{yi},Ai) is a $$PHTS(v)$$PHTS(v) and these $${\mathcal {A}}_i$$Ais form a partition of all cyclic triples and transitive triples on $$Y.$$Y. In this paper, we shall discuss the existence problem of $$OLPHTSs$$OLPHTSs and get the following conclusion: there exists an $$OLPHTS(v)$$OLPHTS(v) if and only if $$v\equiv 0,1$$v?0,1 mod 3 and $$v>3$$v>3.