Shifting lsoperimetric problem: With fixed perimeter, what plane figure has the largest area?
With fixed perimeter, what plane figure has the largest area? Shifting Isoperimetric problem:
Shifting lsoperimetric problem: With fixed area, what plane figure has the smallest perimeter? Steiner symmetrization
With fixed area, what plane figure has the smallest perimeter? Shifting Steiner symmetrization Isoperimetric problem:
Erdos-Ko-Rado Theorem Let FC(),n≥2k. vS,TeF,SnT≠0→Fl≤（ k-1 induction on n F1={S∈F|n∈S Fo={S∈F|m主S} F1={S\{n|S∈F1} F() I.H. FH(-) intersecting 1Fo≤(-) intersecting? F引≤(-2》 1F=Fol+F=1Fol+1F引≤(-)+(-)=(-)
Let F [n] k ⇥ , n ⇥ 2k. |F| ⇥ n 1 k 1 ⇥ Erdős-Ko-Rado Theorem ⇤S, T F, S ⌃ T ⇥= ⌅ induction on n F0 = {S F | n ⇥ S} F1 = {S F | n S} |F| = |F0| + |F1| F 1 = {S \ {n} | S F1} = |F0| + |F 1| F0 [n1] k ⇥ F⇥ 1 [n1] k1 ⇥ intersecting I.H. |F0| n2 k1 ⇥ |F⇥ 1| n2 k2 ⇥ intersecting? I.H. n2 k1 ⇥ + n2 k2 ⇥ = n1 k1 ⇥
Shifting (compression) speckal(） F remains intersecting after deleting n n
Shifting (compression) special F [n] k ⇥ F remains intersecting after deleting n n i
Shifting (compression) FC2n for1≤i<j≤m VTEF,write Tij=(T)Uti (i,)-shift::S（) T∈F, -任 ifj∈T,i年T,and Tij年F, otherwise. S(F)={S(T)|T∈F}
Shifting (compression) Sij (T) = Tij if j T,i ⇥ T, and Tij ⇥ F, T otherwise. F 2[n] for 1 i<j n ⇥T F, write Tij = (T \ {j}) ⌅ {i} (i, j)-shift: Sij (·) ⇥T F, Sij (F) = {Sij (T) | T F}