TY - JOUR
T1 - New turan exponents for two extermal hypergraph problems
AU - SHANGGUAN, CHONG
AU - TAMO, ITZHAK
N1 - Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.
PY - 2020
Y1 - 2020
N2 - An r-uniform hypergraph is called t-cancellative if for any t + 2 distinct edges A1, . . . ,At,B,C, it holds that (∪ ti =1Ai) ∪ B not = (∪ t i=1Ai) ∪ C. It is called t-union-free if for any two distinct subsets scrA; , scrB , each consisting of at most t edges, it holds that ∪ Ain scrA; A not = ∪ Bin scrB B. Let Ct(n, r) (resp., Ut(n, r)) denote the maximum number of edges of a t-cancellative (resp., t-union-free) r-uniform hypergraph on n vertices. Among other results, we show that for fixed r geq 3, t geq 3 and n rightarrow infty , ω (nlfloor 2r t+2 rfloor; +2r ({m}{o}{d} t+2) t+1 ) = Ct(n, r) = O(n lceil r lfloor t/2⌋ +1 ⌉ ) and Ω (n r t 1 ) = Ut(n, r) = O(nlceil r t 1 ⌉ ), thereby significantly narrowing the gap between the previously known lower and upper bounds. In particular, we determine the Turán exponent of Ct(n, r) when 2 | t and (t/2 + 1) | r, and of Ut(n, r) when (t 1) | r. The main tool used in proving the two lower bounds is a novel connection between these problems and sparse hypergraphs.
AB - An r-uniform hypergraph is called t-cancellative if for any t + 2 distinct edges A1, . . . ,At,B,C, it holds that (∪ ti =1Ai) ∪ B not = (∪ t i=1Ai) ∪ C. It is called t-union-free if for any two distinct subsets scrA; , scrB , each consisting of at most t edges, it holds that ∪ Ain scrA; A not = ∪ Bin scrB B. Let Ct(n, r) (resp., Ut(n, r)) denote the maximum number of edges of a t-cancellative (resp., t-union-free) r-uniform hypergraph on n vertices. Among other results, we show that for fixed r geq 3, t geq 3 and n rightarrow infty , ω (nlfloor 2r t+2 rfloor; +2r ({m}{o}{d} t+2) t+1 ) = Ct(n, r) = O(n lceil r lfloor t/2⌋ +1 ⌉ ) and Ω (n r t 1 ) = Ut(n, r) = O(nlceil r t 1 ⌉ ), thereby significantly narrowing the gap between the previously known lower and upper bounds. In particular, we determine the Turán exponent of Ct(n, r) when 2 | t and (t/2 + 1) | r, and of Ut(n, r) when (t 1) | r. The main tool used in proving the two lower bounds is a novel connection between these problems and sparse hypergraphs.
KW - Cancellative hypergraphs
KW - Hypergraph Turán-type problems
KW - Sparse hypergraphs
KW - Union-free hypergraphs
UR - http://www.scopus.com/inward/record.url?scp=85096934464&partnerID=8YFLogxK
U2 - 10.1137/20M1325769
DO - 10.1137/20M1325769
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:85096934464
SN - 0895-4801
VL - 34
SP - 2338
EP - 2345
JO - SIAM Journal on Discrete Mathematics
JF - SIAM Journal on Discrete Mathematics
IS - 4
ER -