Decompositions of Hypergraphs Felix Joos July 2020 Old results - - PowerPoint PPT Presentation

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Decompositions of Hypergraphs Felix Joos July 2020 Old results - - PowerPoint PPT Presentation

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Decompositions of Hypergraphs Felix Joos July 2020 Old results Theorem (Walecki, 1892) K 2 n +1 has a decomposition into edge-disjoint Hamilton cycles. Old results Theorem (Walecki, 1892) K 2 n +1 has a decomposition into edge-disjoint

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SLIDE 1 Decompositions of Hypergraphs Felix Joos July 2020
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SLIDE 2 Old results Theorem (Walecki, 1892) K2n+1 has a decomposition into edge-disjoint Hamilton cycles.
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SLIDE 3 Old results Theorem (Walecki, 1892) K2n+1 has a decomposition into edge-disjoint Hamilton cycles. Theorem (Dirac, 1952) Every graph G on n ≥ 3 vertices with δ(G) ≥ n 2 contains a Hamilton cycle.
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SLIDE 4 Old results Theorem (Walecki, 1892) K2n+1 has a decomposition into edge-disjoint Hamilton cycles. Theorem (Dirac, 1952) Every graph G on n ≥ 3 vertices with δ(G) ≥ n 2 contains a Hamilton cycle. Theorem (Hajnal, Szemer´ edi, 1970) k|n and δ(G) ≥ k−1 k n, then G contains a Kk-factor.
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SLIDE 5 Generalizations Theorem (Csaba, K¨ uhn, Lo, Osthus, Treglown, 2016) G r-regular with r ≥ n 2 and even, n large, then G has a Hamilton decomposition.
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SLIDE 6 Generalizations Theorem (Csaba, K¨ uhn, Lo, Osthus, Treglown, 2016) G r-regular with r ≥ n 2 and even, n large, then G has a Hamilton decomposition. Theorem (B¨
  • ttcher, Schacht, Taraz, 2009)
χ(H) = k, H has bandwidth o(n) and ∆(H) = O(1) δ(G) ≥ ( k−1 k +o(1))n, then H ⊂ G.
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SLIDE 7 Open problems Conjecture (Nash-Williams, 1970) δ(G) ≥ 3n 4 and G is triangle-divisible, then G has a triangle decomposition
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SLIDE 8 Open problems Conjecture (Nash-Williams, 1970) δ(G) ≥ 3n 4 and G is triangle-divisible, then G has a triangle decomposition Problem Given a graph H, determine δH where δH is the least δ such that for every ε > 0 and G on n (large) vertices with δ(G) ≥ (δ +ε)n has a H-decomposition subject to divisibility conditions?
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SLIDE 9 A step forward Fractional H-decomposition of G: ω: {copies of H in G} → [0,1] such that H∋e ω(H) = 1 for e ∈ E(G).
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SLIDE 10 A step forward Fractional H-decomposition of G: ω: {copies of H in G} → [0,1] such that H∋e ω(H) = 1 for e ∈ E(G). δ∗ H fractional version of δH.
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SLIDE 11 A step forward Fractional H-decomposition of G: ω: {copies of H in G} → [0,1] such that H∋e ω(H) = 1 for e ∈ E(G). δ∗ H fractional version of δH. Theorem (Barber, K¨ uhn, Lo, Osthus, 2016; Glock, K¨ uhn, Lo, Montgomery, Osthus, 2019) δH ∈ {δ∗ H,1− 1 χ,1− 1 χ+1} for χ = χ(H) ≥ 5 solved for bipartite H General tool: turning fractional decompositions into decompositions
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SLIDE 12 Cycles Theorem (Barber, K¨ uhn, Lo, Osthus, 2016) δC4 = 2 3 and δC2ℓ = 1 2 for ℓ ≥ 3 δC2ℓ+1 = δ∗ C2ℓ+1
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SLIDE 13 Cycles Theorem (Barber, K¨ uhn, Lo, Osthus, 2016) δC4 = 2 3 and δC2ℓ = 1 2 for ℓ ≥ 3 δC2ℓ+1 = δ∗ C2ℓ+1 Theorem (J., M. K¨ uhn, 2020+) δC2ℓ+1 → 1 2 (ℓ → ∞)
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SLIDE 14 Hypergraphs - old results G k-uniform (k-graph): edges of size k dm(S) = number of edges containing S for |S| = m δm(G) = minS dm(S) δ(G) = δk−1(G)
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SLIDE 15 Hypergraphs - old results G k-uniform (k-graph): edges of size k dm(S) = number of edges containing S for |S| = m δm(G) = minS dm(S) δ(G) = δk−1(G) tight cycle = cyclic vertex ordering, all k consecutive vertices form an edge
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SLIDE 16 Hypergraphs - old results G k-uniform (k-graph): edges of size k dm(S) = number of edges containing S for |S| = m δm(G) = minS dm(S) δ(G) = δk−1(G) tight cycle = cyclic vertex ordering, all k consecutive vertices form an edge Theorem (R¨
  • dl, Ruci´
nski, Szemer´ edi, 2008) δ(G) ≥ ( 1 2 +o(1))n, then G contains a (tight) Hamilton cycle
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SLIDE 17 Hypergraphs - old results G k-uniform (k-graph): edges of size k dm(S) = number of edges containing S for |S| = m δm(G) = minS dm(S) δ(G) = δk−1(G) tight cycle = cyclic vertex ordering, all k consecutive vertices form an edge Theorem (R¨
  • dl, Ruci´
nski, Szemer´ edi, 2008) δ(G) ≥ ( 1 2 +o(1))n, then G contains a (tight) Hamilton cycle Theorem (Lang, Sahueza-Matamala; Polcyn, Reiher, R¨
  • dl,
Sch¨ ulke, 2020+) δk−2(G) ≥ ( 5 9 +o(1))n2/2, then G contains a (tight) Hamilton cycle
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SLIDE 18 New results
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SLIDE 19 New results Theorem (J., K¨ uhn, 2020+) δ∗ C(k) ℓ → 1 2 for ℓ → ∞
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SLIDE 20 New results Theorem (J., K¨ uhn, 2020+) δ∗ C(k) ℓ → 1 2 for ℓ → ∞ Proof method: Restriction systems + random walks
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SLIDE 21 Proof Sketch 1- Simple random walk
  • n
a reg
  • la
, non - bipahte , counected graph ⇒ Uniform limit distribution Z Find a d- regulaer subgraph in the line graph der > & : dd" "" "bes reshiction system 3- Markov chain
  • n
  • rdered
edges ( avoid walking around a rete) 4- P [ traue to Fish steps ] =p ⇐> # b- walks from ötof =p d " h h conform with the restriktion system 5 pm → 1- ( very quichly in b)
  • ↳x
Zeh) Kind .
  • f
e and f!
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SLIDE 22 New results II
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SLIDE 23 New results II Theorem (R¨
  • dl, Ruci´
nski, Szemer´ edi, 2008) ∀ǫ > 0,k ∈ N the following holds for all large n: G k-graph with δ(G) ≥ ( 1 2 +ε)n, then G contains a (tight) Hamilton cycle
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SLIDE 24 New results II Theorem (R¨
  • dl, Ruci´
nski, Szemer´ edi, 2008) ∀ǫ > 0,k ∈ N the following holds for all large n: G k-graph with δ(G) ≥ ( 1 2 +ε)n, then G contains a (tight) Hamilton cycle Theorem (J., K¨ uhn, Sch¨ ulke, 2020+) ∀ǫ > 0,k ∈ N ∃ ǫ′ > 0 such that for all large n:
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SLIDE 25 New results II Theorem (R¨
  • dl, Ruci´
nski, Szemer´ edi, 2008) ∀ǫ > 0,k ∈ N the following holds for all large n: G k-graph with δ(G) ≥ ( 1 2 +ε)n, then G contains a (tight) Hamilton cycle Theorem (J., K¨ uhn, Sch¨ ulke, 2020+) ∀ǫ > 0,k ∈ N ∃ ǫ′ > 0 such that for all large n:
  • G k-graph with δ(G) ≥ ( 1
2 +ǫ)n
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SLIDE 26 New results II Theorem (R¨
  • dl, Ruci´
nski, Szemer´ edi, 2008) ∀ǫ > 0,k ∈ N the following holds for all large n: G k-graph with δ(G) ≥ ( 1 2 +ε)n, then G contains a (tight) Hamilton cycle Theorem (J., K¨ uhn, Sch¨ ulke, 2020+) ∀ǫ > 0,k ∈ N ∃ ǫ′ > 0 such that for all large n:
  • G k-graph with δ(G) ≥ ( 1
2 +ǫ)n
  • |d1(v)−d1(u)| ≤ ǫ′nk−1
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SLIDE 27 New results II Theorem (R¨
  • dl, Ruci´
nski, Szemer´ edi, 2008) ∀ǫ > 0,k ∈ N the following holds for all large n: G k-graph with δ(G) ≥ ( 1 2 +ε)n, then G contains a (tight) Hamilton cycle Theorem (J., K¨ uhn, Sch¨ ulke, 2020+) ∀ǫ > 0,k ∈ N ∃ ǫ′ > 0 such that for all large n:
  • G k-graph with δ(G) ≥ ( 1
2 +ǫ)n
  • |d1(v)−d1(u)| ≤ ǫ′nk−1
then G contains (e(G)−ǫnk)/n edge-disjoint Hamilton cycles.
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SLIDE 28 New results II Theorem (J., K¨ uhn, Sch¨ ulke, 2020+) ∀ǫ > 0,k ∈ N ∃ ǫ′ > 0 such that for all large n:
  • G k-graph with δ(G) ≥ ( 1
2 +ǫ)n
  • |d1(v)−d1(u)| ≤ ǫ′nk−1
then G contains (e(G)−ǫnk)/n edge-disjoint Hamilton cycles.
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SLIDE 29 New results II Theorem (J., K¨ uhn, Sch¨ ulke, 2020+) ∀ǫ > 0,k ∈ N ∃ ǫ′ > 0 such that for all large n:
  • G k-graph with δ(G) ≥ ( 1
2 +ǫ)n
  • |d1(v)−d1(u)| ≤ ǫ′nk−1
then G contains (e(G)−ǫnk)/n edge-disjoint Hamilton cycles. Corollary Vertex-regular k-graphs G with δ(G) ≥ ( 1 2 +o(1))n can be approximately decomposed into Hamilton cycles with an arbitrary good precision.
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SLIDE 30 New results II Theorem (J., K¨ uhn, Sch¨ ulke, 2020+) ∀ǫ > 0,k ∈ N ∃ ǫ′ > 0 such that for all large n:
  • G k-graph with δ(G) ≥ ( 1
2 +ǫ)n
  • |d1(v)−d1(u)| ≤ ǫ′nk−1
then G contains (e(G)−ǫnk)/n edge-disjoint Hamilton cycles. Corollary Vertex-regular k-graphs G with δ(G) ≥ ( 1 2 +o(1))n can be approximately decomposed into Hamilton cycles with an arbitrary good precision. Proof method: δ∗ Cℓ ≤ 1 2 +ǫ for large enough ℓ; random process; absorption
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SLIDE 31 Summary I
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SLIDE 32 Summary I
  • Fractional ℓ-cycle decompositions
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SLIDE 33 Summary I
  • Fractional ℓ-cycle decompositions
  • approx. decompositions into Hamilton cycles
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SLIDE 34 Summary I
  • Fractional ℓ-cycle decompositions
  • approx. decompositions into Hamilton cycles
in hypergraphs under very weak assumptions
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SLIDE 35 Graph decompositions Three conjectures: ◮ Ringel: K2n+1 into any tree with n edges ◮ Tree packing conj.: Kn into trees T1,...,Tn−1 with e(Ti) = i ◮ Oberwolfach problem: K2n+1 into any spanning union of cycles
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SLIDE 36 Graph decompositions - progress Approximate decompositions:
  • ∆ = O(1), trees, almost spanning, Kn B¨
  • ttcher, Hladk´
y, Piguet, Taraz, 16
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SLIDE 37 Graph decompositions - progress Approximate decompositions:
  • ∆ = O(1), trees, almost spanning, Kn B¨
  • ttcher, Hladk´
y, Piguet, Taraz, 16
  • ∆ = O(1), separable, almost spanning, Kn Messuti, R¨
  • dl, Schacht, 17
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SLIDE 38 Graph decompositions - progress Approximate decompositions:
  • ∆ = O(1), trees, almost spanning, Kn B¨
  • ttcher, Hladk´
y, Piguet, Taraz, 16
  • ∆ = O(1), separable, almost spanning, Kn Messuti, R¨
  • dl, Schacht, 17
  • ∆ = O(1), separable, spanning, Kn Ferber, Lee, Mousset, 16
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SLIDE 39 Graph decompositions - progress Approximate decompositions:
  • ∆ = O(1), trees, almost spanning, Kn B¨
  • ttcher, Hladk´
y, Piguet, Taraz, 16
  • ∆ = O(1), separable, almost spanning, Kn Messuti, R¨
  • dl, Schacht, 17
  • ∆ = O(1), separable, spanning, Kn Ferber, Lee, Mousset, 16
  • ∆ = O(1), spanning, multipart., quasiran. Kim, K¨
uhn, Osthus, Tyomkyn, 19
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SLIDE 40 Graph decompositions - progress Approximate decompositions:
  • ∆ = O(1), trees, almost spanning, Kn B¨
  • ttcher, Hladk´
y, Piguet, Taraz, 16
  • ∆ = O(1), separable, almost spanning, Kn Messuti, R¨
  • dl, Schacht, 17
  • ∆ = O(1), separable, spanning, Kn Ferber, Lee, Mousset, 16
  • ∆ = O(1), spanning, multipart., quasiran. Kim, K¨
uhn, Osthus, Tyomkyn, 19
  • ∆ ≤ polypn, trees, spanning, Gn,p Ferber, Samotij, 18+
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SLIDE 41 Graph decompositions - progress Approximate decompositions:
  • ∆ = O(1), trees, almost spanning, Kn B¨
  • ttcher, Hladk´
y, Piguet, Taraz, 16
  • ∆ = O(1), separable, almost spanning, Kn Messuti, R¨
  • dl, Schacht, 17
  • ∆ = O(1), separable, spanning, Kn Ferber, Lee, Mousset, 16
  • ∆ = O(1), spanning, multipart., quasiran. Kim, K¨
uhn, Osthus, Tyomkyn, 19
  • ∆ ≤ polypn, trees, spanning, Gn,p Ferber, Samotij, 18+
  • ∆ ≤ n/logn, bo. dege., spanning, quasiran. Allen, B¨
  • ttcher, Hladk´
y, Piguet, 19
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SLIDE 42 Graph decompositions - progress Approximate decompositions:
  • ∆ = O(1), trees, almost spanning, Kn B¨
  • ttcher, Hladk´
y, Piguet, Taraz, 16
  • ∆ = O(1), separable, almost spanning, Kn Messuti, R¨
  • dl, Schacht, 17
  • ∆ = O(1), separable, spanning, Kn Ferber, Lee, Mousset, 16
  • ∆ = O(1), spanning, multipart., quasiran. Kim, K¨
uhn, Osthus, Tyomkyn, 19
  • ∆ ≤ polypn, trees, spanning, Gn,p Ferber, Samotij, 18+
  • ∆ ≤ n/logn, bo. dege., spanning, quasiran. Allen, B¨
  • ttcher, Hladk´
y, Piguet, 19 Decompositions:
  • ∆ = O(1), trees, (almost) spanning, quasiran. J., Kim, K¨
uhn, Osthus, 19
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SLIDE 43 Graph decompositions - progress Approximate decompositions:
  • ∆ = O(1), trees, almost spanning, Kn B¨
  • ttcher, Hladk´
y, Piguet, Taraz, 16
  • ∆ = O(1), separable, almost spanning, Kn Messuti, R¨
  • dl, Schacht, 17
  • ∆ = O(1), separable, spanning, Kn Ferber, Lee, Mousset, 16
  • ∆ = O(1), spanning, multipart., quasiran. Kim, K¨
uhn, Osthus, Tyomkyn, 19
  • ∆ ≤ polypn, trees, spanning, Gn,p Ferber, Samotij, 18+
  • ∆ ≤ n/logn, bo. dege., spanning, quasiran. Allen, B¨
  • ttcher, Hladk´
y, Piguet, 19 Decompositions:
  • ∆ = O(1), trees, (almost) spanning, quasiran. J., Kim, K¨
uhn, Osthus, 19
  • Oberwolfach problem Glock, J., Kim, K¨
uhn, Osthus, 18+
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SLIDE 44 Graph decompositions - progress Approximate decompositions:
  • ∆ = O(1), trees, almost spanning, Kn B¨
  • ttcher, Hladk´
y, Piguet, Taraz, 16
  • ∆ = O(1), separable, almost spanning, Kn Messuti, R¨
  • dl, Schacht, 17
  • ∆ = O(1), separable, spanning, Kn Ferber, Lee, Mousset, 16
  • ∆ = O(1), spanning, multipart., quasiran. Kim, K¨
uhn, Osthus, Tyomkyn, 19
  • ∆ ≤ polypn, trees, spanning, Gn,p Ferber, Samotij, 18+
  • ∆ ≤ n/logn, bo. dege., spanning, quasiran. Allen, B¨
  • ttcher, Hladk´
y, Piguet, 19 Decompositions:
  • ∆ = O(1), trees, (almost) spanning, quasiran. J., Kim, K¨
uhn, Osthus, 19
  • Oberwolfach problem Glock, J., Kim, K¨
uhn, Osthus, 18+
  • as ABHP + many leaves Allen, B¨
  • ttcher, Clemens, Taraz, 19+
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SLIDE 45 Graph decompositions - progress Approximate decompositions:
  • ∆ = O(1), trees, almost spanning, Kn B¨
  • ttcher, Hladk´
y, Piguet, Taraz, 16
  • ∆ = O(1), separable, almost spanning, Kn Messuti, R¨
  • dl, Schacht, 17
  • ∆ = O(1), separable, spanning, Kn Ferber, Lee, Mousset, 16
  • ∆ = O(1), spanning, multipart., quasiran. Kim, K¨
uhn, Osthus, Tyomkyn, 19
  • ∆ ≤ polypn, trees, spanning, Gn,p Ferber, Samotij, 18+
  • ∆ ≤ n/logn, bo. dege., spanning, quasiran. Allen, B¨
  • ttcher, Hladk´
y, Piguet, 19 Decompositions:
  • ∆ = O(1), trees, (almost) spanning, quasiran. J., Kim, K¨
uhn, Osthus, 19
  • Oberwolfach problem Glock, J., Kim, K¨
uhn, Osthus, 18+
  • as ABHP + many leaves Allen, B¨
  • ttcher, Clemens, Taraz, 19+
  • Ringel’s conjecture Pokrovskiy, Montgomery, Sudakov, 20+
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SLIDE 46 Graph decompositions - progress Approximate decompositions:
  • ∆ = O(1), trees, almost spanning, Kn B¨
  • ttcher, Hladk´
y, Piguet, Taraz, 16
  • ∆ = O(1), separable, almost spanning, Kn Messuti, R¨
  • dl, Schacht, 17
  • ∆ = O(1), separable, spanning, Kn Ferber, Lee, Mousset, 16
  • ∆ = O(1), spanning, multipart., quasiran. Kim, K¨
uhn, Osthus, Tyomkyn, 19
  • ∆ ≤ polypn, trees, spanning, Gn,p Ferber, Samotij, 18+
  • ∆ ≤ n/logn, bo. dege., spanning, quasiran. Allen, B¨
  • ttcher, Hladk´
y, Piguet, 19 Decompositions:
  • ∆ = O(1), trees, (almost) spanning, quasiran. J., Kim, K¨
uhn, Osthus, 19
  • Oberwolfach problem Glock, J., Kim, K¨
uhn, Osthus, 18+
  • as ABHP + many leaves Allen, B¨
  • ttcher, Clemens, Taraz, 19+
  • Ringel’s conjecture Pokrovskiy, Montgomery, Sudakov, 20+
  • 2-factors, quasiran. Keevash, Staden, 20+
slide-47
SLIDE 47 Graph decompositions - progress Approximate decompositions:
  • ∆ = O(1), trees, almost spanning, Kn B¨
  • ttcher, Hladk´
y, Piguet, Taraz, 16
  • ∆ = O(1), separable, almost spanning, Kn Messuti, R¨
  • dl, Schacht, 17
  • ∆ = O(1), separable, spanning, Kn Ferber, Lee, Mousset, 16
  • ∆ = O(1), spanning, multipart., quasiran. Kim, K¨
uhn, Osthus, Tyomkyn, 19
  • ∆ ≤ polypn, trees, spanning, Gn,p Ferber, Samotij, 18+
  • ∆ ≤ n/logn, bo. dege., spanning, quasiran. Allen, B¨
  • ttcher, Hladk´
y, Piguet, 19 Decompositions:
  • ∆ = O(1), trees, (almost) spanning, quasiran. J., Kim, K¨
uhn, Osthus, 19
  • Oberwolfach problem Glock, J., Kim, K¨
uhn, Osthus, 18+
  • as ABHP + many leaves Allen, B¨
  • ttcher, Clemens, Taraz, 19+
  • Ringel’s conjecture Pokrovskiy, Montgomery, Sudakov, 20+
  • 2-factors, quasiran. Keevash, Staden, 20+
  • Ringel’s conjecture, quasiran. Keevash, Staden, 20+
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SLIDE 48 Hypergraph decompositions - progress
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SLIDE 49 Hypergraph decompositions - progress Existence conjecture: Keevash, 14+
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SLIDE 50 Hypergraph decompositions - progress Existence conjecture: Keevash, 14+ Alternative proof + extensions: Glock, K¨ uhn, Lo, Osthus, 17+
slide-51
SLIDE 51 Hypergraph decompositions - progress Existence conjecture: Keevash, 14+ Alternative proof + extensions: Glock, K¨ uhn, Lo, Osthus, 17+ Multipartite setting: Keevash, 18+
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SLIDE 52 Hypergraph decompositions - progress Existence conjecture: Keevash, 14+ Alternative proof + extensions: Glock, K¨ uhn, Lo, Osthus, 17+ Multipartite setting: Keevash, 18+ Quasirandom hypergraphs: ǫ > 0, t ∈ N, d ∈ (0,1] and suppose G has n vertices. G is (ǫ,t,d)-typical if
  • S∈S
NG(S)
  • = (1±ǫ)d|S|n
for all sets S of (k −1)-sets of V (G) with |S| ≤ t.
slide-53
SLIDE 53 New results III Theorem (Ehard, J., 2020+) ∀k,α,d0 > 0 ∃ n0,t ∈ N,ε > 0:
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SLIDE 54 New results III Theorem (Ehard, J., 2020+) ∀k,α,d0 > 0 ∃ n0,t ∈ N,ε > 0: G k-graph, n ≥ n0 vertices, (ε,t,d)-typical with d ≥ d0
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SLIDE 55 New results III Theorem (Ehard, J., 2020+) ∀k,α,d0 > 0 ∃ n0,t ∈ N,ε > 0: G k-graph, n ≥ n0 vertices, (ε,t,d)-typical with d ≥ d0 H1,...,Hℓ k-graphs, n vertices each, ∆1(Hi) ≤ α−1 and i∈[ℓ] e(Hi) ≤ (1−α)e(G).
slide-56
SLIDE 56 New results III Theorem (Ehard, J., 2020+) ∀k,α,d0 > 0 ∃ n0,t ∈ N,ε > 0: G k-graph, n ≥ n0 vertices, (ε,t,d)-typical with d ≥ d0 H1,...,Hℓ k-graphs, n vertices each, ∆1(Hi) ≤ α−1 and i∈[ℓ] e(Hi) ≤ (1−α)e(G). Then G contains H1,...,Hℓ as edge-disjoint subgraphs.
slide-57
SLIDE 57 Multipartite hypergraphs (foränrdiäty) X 4) < ä "

±

. KR):{ I. - er} ¢ A)Rki '

|

Ht : Nil
  • Hehn

viykt.cl/-typical.r..iosn

a .!üÄ EKHN" @

@

slide-58
SLIDE 58 New results IV Theorem (Ehard, J., 2020+)
  • Approx. decomp. of quasirandom multipartite k-graphs
into bounded degree k-graphs with the same multipartite structure.
slide-59
SLIDE 59 New results IV Theorem (Ehard, J., 2020+)
  • Approx. decomp. of quasirandom multipartite k-graphs
into bounded degree k-graphs with the same multipartite structure. Hypergraph blow-up lemma for approximate decompositions for quasirandom k-graphs
slide-60
SLIDE 60 New results IV Theorem (Ehard, J., 2020+)
  • Approx. decomp. of quasirandom multipartite k-graphs
into bounded degree k-graphs with the same multipartite structure. Hypergraph blow-up lemma for approximate decompositions for quasirandom k-graphs Asked by Keevash and Kim, K¨ uhn, Osthus, Tyomkyn
slide-61
SLIDE 61 New results IV Theorem (Ehard, J., 2020+)
  • Approx. decomp. of quasirandom multipartite k-graphs
into bounded degree k-graphs with the same multipartite structure. Hypergraph blow-up lemma for approximate decompositions for quasirandom k-graphs Asked by Keevash and Kim, K¨ uhn, Osthus, Tyomkyn More features: The packing itself exhibits strong quasirandom properties which is very useful for applications
slide-62
SLIDE 62 Quasirandom properties I H , Hz 43

④ ④ ⑦

tot .

i

'
slide-63
SLIDE 63 Quasirandom properties II H , Hz " t.fevcHP-sio.is

to

£ Constndro such that ÷ :*::*:

"

slide-64
SLIDE 64 Proof ideas Multipartite setting implies the other setting
slide-65
SLIDE 65 Proof ideas Multipartite setting implies the other setting
  • 1. Proceed cluster by cluster:
iteratively embed almost all vertices in i∈[ℓ] X Hi j into Vj
slide-66
SLIDE 66 Proof ideas Multipartite setting implies the other setting
  • 1. Proceed cluster by cluster:
iteratively embed almost all vertices in i∈[ℓ] X Hi j into Vj
  • 2. Complete the embedding using an extra edge slice
slide-67
SLIDE 67 Proof ideas II H1 X H1 1 X H1 2 X H1 3 G V1 V2 V3 H2 X H2 1 X H2 2 X H2 3 g
slide-68
SLIDE 68 Proof ideas III

iö

pachings - matchings
slide-69
SLIDE 69 Summary II
  • Approx. decompositions of quasirandom k-graphs
in the normal and multipartite setting into bounded degree k-graphs
slide-70
SLIDE 70 Applications Consider a hypergraph as a simplicial complex: Hamilton cycle in a k-graph = spanning Sk−1
slide-71
SLIDE 71 Applications Consider a hypergraph as a simplicial complex: Hamilton cycle in a k-graph = spanning Sk−1 Georgakopoulos, Haslegrave, Narayanan, Montgomery, 18+: 3-graph G with δ(G) ≥ ( 1 3 +o(1))n, then G contains a spanning S2
slide-72
SLIDE 72 Applications Consider a hypergraph as a simplicial complex: Hamilton cycle in a k-graph = spanning Sk−1 Georgakopoulos, Haslegrave, Narayanan, Montgomery, 18+: 3-graph G with δ(G) ≥ ( 1 3 +o(1))n, then G contains a spanning S2 Ehard, J. 20+: Typical 3-graphs can be approx. decomposed into spanning S2
slide-73
SLIDE 73 Applications Consider a hypergraph as a simplicial complex: Hamilton cycle in a k-graph = spanning Sk−1 Georgakopoulos, Haslegrave, Narayanan, Montgomery, 18+: 3-graph G with δ(G) ≥ ( 1 3 +o(1))n, then G contains a spanning S2 Ehard, J. 20+: Typical k-graphs can be approx. decomposed into spanning Sk−1