Singmaster’s conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal’s triangle; that is, for any natural number $t \geq 2$, the number of solutions to the equation $\binom{n}{m} = t$ for natural numbers $1 \leq m \lt n$ is bounded. In this paper we establish this result in the interior region $\exp(\log^{2/3+\varepsilon} n) \leq m \leq n  \exp(\log^{2/3+\varepsilon} n)$ for any fixed ɛ > 0. Indeed, when t is sufficiently large depending on ɛ, we show that there are at most four solutions (or at most two in either half of Pascal’s triangle) in this region. We also establish analogous results for the equation $(n)_m = t$, where $(n)_m := n(n1) \dots (nm+1)$ denotes the falling factorial.
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Turán graphs with bounded matching number
We determine the maximum possible number of edges of a graph with n vertices, matching number at most s and clique number at most k for all admissible values of the parameters.
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 Award ID(s):
 2154082
 NSFPAR ID:
 10498656
 Publisher / Repository:
 Elsevier
 Date Published:
 Journal Name:
 Journal of Combinatorial Theory, Series B
 Volume:
 165
 Issue:
 C
 ISSN:
 00958956
 Page Range / eLocation ID:
 223 to 229
 Subject(s) / Keyword(s):
 Combinatorics Discrete Mathematics
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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