Turán numbers of vertex-disjoint cliques in r-partite graphs

Refuting conjectures in extremal combinatorics via linear programming

“…Solving this IP directly gives the following construction for (n, k, s) = (11,5,3) This gives k(7, 4) = 120 and hence in order to disprove Conjecture 3.18 our goal is to show k(9, 4) ≥ 481. One can formulate this problem as an IP as follows.…”

“…De Silva et al [11] observed that the graph ((n 1 + n 2 − k + 1)K 1 ∪ K k−1,n 3 ) + K 4 does not contain kK 3 , hence ex(K n 1 ,n 2 ,n 3 ,n 4 , kK 3 ) ≥ (n 1 + n 2 + n 3 )n 4 + (k − 1)n 3 .…”

Weighted Turán problems with applications

“…In this section we address a problem suggested by De Silva, Heysse, Kapilow, Schenfisch, and Young [20]. For general graphs H, F we define the extremal number of F with host graph H, ex(H, F ), to be the largest number of edges in any graph G such that F ⊆ G ⊆ H. Of course when H = K n we just get the standard extremal number ex(n, F ).…”

“…This variation originated with Zarankiewicz, who was interested in the case where the host graph is bipartite, specifically ex(K n,n , K s,s ), [23]. More recently, De Silva, Heysse, Kapilow, Schenfisch and Young determined the numbers ex(K a 1 ,a 2 ,...,a ℓ , sK ℓ ), forbidding a union of disjoint cliques in a complete multi-partite host graph, [20]. The authors suggest an open problem; determining ex(K k 1 ,k 2 ,...,kr , sK ℓ ), where the number of partite sets in the host graph k ≥ r, the size of the forbidden cliques.…”

Turán Problems for Vertex-disjoint Cliques in Multi-partite Hypergraphs

“…One of the approaches to the celebrated Szemerédi Theorem [15,16,21,22] is to determine the maximum number of edges in a special (s + 1)partite s-graph, in which each edge is contained in exactly one copy of K (s) s+1 . Recently, Turán problems in multi-partite graphs have received a lot of attention, see [3,6,7,17]. In [6], De Silva, Heysse, and Young determined ex(K n 1 ,...,nr , kK 2 ).…”